Regularity Of Matrices Research Articles

A bridging domain and strain computation method for coupled atomistic–continuum modelling of solids

AbstractWe present a multiscale method that couples atomistic models with continuum mechanics. The method is based on an overlapping domain‐decomposition scheme. Constraints are imposed by a Lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap. An efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material. An iterative scheme is utilized to optimize the coupled configuration. Conditions for the regularity of the constrained matrices are determined. A method for computing strain in atomistic models and handshake domains is formulated based on a moving least‐square approximation which includes both extensional and angle‐bending terms. It is shown that this method exactly computes the linear strain field. Applications to the fracture of defected single‐layer atomic sheets and nanotubes are given. Copyright © 2006 John Wiley & Sons, Ltd.

Regularity of Interval Matrices and Theorems of the Alternatives

We give several characterizations of regularity of interval matrices. All of them have to do with solvability of certain systems of nonlinear equations or inequalities. The most illustrative of them is the following one: an interval matrix [Ac − Δ, Ac + Δ] is regular if and only if the nonlinear inequality \(|x| > \Delta |A_{c}^{-1}x|\) has a solution in each orthant. These results are then applied to derive two theorems of the alternatives for inequalities with absolute values.

A Theorem on Regularity of Interval Matrices

Abstract. We give a theorem which enables the determination of whether an interval matrix is regular or not.

How Strong Is Strong Regularity?

We compare the notions of regularity and strong regularity of interval matrices. For an n × n interval matrix A we construct 2 n open convex cones, all of them lying in the interior of the nonnegative orthant. It is shown that regularity of A is characterized by nonemptiness of all these cones, whereas strong regularity is characterized by nonemptiness of their intersection.

ADAPTIVE FEATURE SPACES FOR LAND COVER CLASSIFICATION WITH LIMITED GROUND TRUTH DATA

Classification of land cover based on hyperspectral data is very challenging because typically tens of classes with uneven priors are involved, the inputs are high dimensional, and there is often scarcity of labeled data. Several researchers have observed that it is often preferable to decompose a multiclass problem into multiple two-class problems, solve each such subproblem using a suitable binary classifier, and then combine the outputs of this collection of classifiers in a suitable manner to obtain the answer to the original multiclass problem. This approach is taken by the popular error correcting output codes (ECOC) technique, as well by the binary hierarchical classifier (BHC). Classical techniques for dealing with small sample sizes include regularization of covariance matrices and feature reduction. In this paper we address the twin problems of small sample sizes and multiclass settings by proposing a feature reduction scheme that adaptively adjusts to the amount of labeled data available. This scheme can be used in conjunction with ECOC and the BHC, as well as other approaches such as round-robin classification that decompose a multiclass problem into a number of two (meta)-class problems. In particular, we develop the best-basis binary hierarchical classifier (BB-BHC) and best basis ECOC (BB-ECOC) families of models that are adapted to "small sample size" situations. Currently, there are few studies that compare the efficacy of different approaches to multiclass problems in general settings as well as in the specific context of small sample sizes. Our experiments on two sets of remote sensing data show that both BB-BHC and BB-ECOC methods are superior to their nonadaptive versions when faced with limited data, with the BB-BHC showing a slight edge in terms of classification accuracy as well as interpretability.

On regularity of block triangular fuzzy matrices

Necessary and sufficient conditions are given for the regularity of block triangular fuzzy matrices. This leads to characterization of idempotency of a class of triangular Toeplitz matrices. As an application, the existence of group inverse of a block triangular fuzzy matrix is discussed. Equivalent conditions for a regular block triangular fuzzy matrix to be expressed as a sum of regular block fuzzy matrices is derived. Further, fuzzy relational equations consistency is studied.

PERMUTATION GROUP APPROACH TO THE ONE-DIMENSIONAL XXX HEISENBERG OPEN SPIN-1/2 CHAINS

Permutation group approach to the one-dimensional XXX Heisenberg open spin-1/2 chain with nearest neighbor interaction is proposed, which is formulated not only for half-filling excitations, but also for general case. Regularity of energy matrices and the corresponding wavefunctions are made up by using an induction method, which makes it possible to study the eigenvalue problem systematically. A Mathematica package for generating the energy matrices is implemented according to this procedure.

Some remarks on projective-free rings

The homological classification of rings is based on the length of projective resolutions of modules, but it tells us nothing about the projective modules themselves. For information on them we have to turn to the monoid of projectives; thus for any ring R we have P(R), the monoid of (isomorphism classes of) finitely generated projective R-modules, with the operation [P ] + [Q] = [P ⊕ Q]. The simplest form that this monoid can take is P(R) ∼= N, the natural numbers, with [R] corresponding to 1. It means that every finitely generated projective R-module is free, of unique rank. Such a ring is said to be projective-free; in particular, since the rank of any free module is unique, such a ring has invariant basis number (IBN). Our aim in this note is to derive some results on projective-free rings. We shall show in Section 2 that surjective homomorphisms which are local, i.e. map non-units to non-units, do not diminish the monoid of projectives. In particular, we shall obtain conditions ensuring that a ring homomorphism reflects projective-freeness. A slightly weaker conclusion is reached in Section 3 by assuming the regularity of full matrices. I am indebted to G. M. Bergman whose comments on an earlier version greatly improved the presentation; in particular, Lemma 1 and suggestions leading to Theorem 2 are due to him. Throughout all rings are associative, with a unit element 1 which is preserved by homomorphisms and inherited by subrings.

Strong regularity of matrices in general max–min algebra

The problem of the strong regularity of a square matrix in a general max–min algebra is considered and a necessary and sufficient condition using the trapezoidal property is described. The results are valid without any restrictions on the underlying max–min algebra, concerning the density, or the boundedness. Previous results on this topic are special cases of the theorems presented in this paper.

Criteria for the strong regularity of J-inner functions and γ-generating matrices

The class of left and right strongly regular J-inner mvf's plays an important role in bitangential interpolation problems and in bitangential direct and inverse problems for canonical systems of integral and differential equations. A new criterion for membership in this class is presented in terms of the matricial Muckenhoupt condition (A2) that was introduced for other purposes by Treil and Volberg. Analogous results are also obtained for the class of γ-generating functions that intervene in the Nehari problem. The new criterion is simpler than the criterion that we presented earlier. A determinental criterion is also presented.

Max algebra and the linear assignment problem

Max-algebra, where the classical arithmetic operations of addition and multiplication are replaced by a⊕b:=max(a, b) and a⊗b:=a+b offers an attractive way for modelling discrete event systems and optimization problems in production and transportation. Moreover, it shows a strong similarity to classical linear algebra: for instance, it allows a consideration of linear equation systems and the eigenvalue problem. The max-algebraic permanent of a matrix A corresponds to the maximum value of the classical linear assignment problem with cost matrix A. The analogue of van der Waerden's conjecture in max-algebra is proved. Moreover the role of the linear assignment problem in max-algebra is elaborated, in particular with respect to the uniqueness of solutions of linear equation systems, regularity of matrices and the minimal-dimensional realisation of discrete event systems. Further, the eigenvalue problem in max-algebra is discussed. It is intimately related to the best principal submatrix problem which is finally investigated: Given an integer k, 1≤k≤n, find a (k×k) principal submatrix of the given (n×n) matrix which yields among all principal submatrices of the same size the maximum (minimum) value for an assignment. For k=1,2,...,n, the maximum assignment problem values of the principal (k×k) submatrices are the coefficients of the max-algebraic characteristic polynomial of the matrix for A. This problem can be used to model job rotations.

About the von Neumann regularity of triangular block matrices

Necessary and sufficient conditions are given for the von Neumann regularity of triangular block matrices with von Neumann regular diagonal blocks over arbitrary rings. This leads to the characterization of the von Neumann regularity of a class of triangular Toeplitz matrices over arbitrary rings. Some special results and a new algorithm are derived for triangular Toeplitz matrices over commutative rings. Finally, the Drazin invertibility of some companion matrices over arbitrary rings is considered, as an application.

An Algorithm for Checking Regularity of Interval Matrices

Checking regularity (or singularity) of interval matrices is a known NP-hard problem. In this paper a general algorithm for checking regularity/singularity is presented which is not a priori exponential. The algorithm is based on a theoretical result according to which regularity may be judged from any single component of the solution set of an associated system of linear interval equations. Numerical experiments (with interval matrices up to the size n = 50) confirm that this approach brings an essential decrease in the amount of operations needed.

Necessary and sufficient conditions for the regularity of an interval matrix

Verifiable necessary and sufficient conditions for regularity of an interval matrix are given. Based on these conditions an algorithm for checking the regularity of interval matrices is developed. It is shown that this algorithm can be modified to determine the stability of a class of interval matrices.

Elimination of rigid body modes from discretized boundary integral equations

Free rigid body modes in Neumann problems are typically eliminated by suitably restraining the body. An alternative approach, here called “regularization”, involves first computing the singular stiffness matrix and then suitably modifying it using ideas from linear algebra. This idea has been suggested by Vêrchery (1990) for symmetric matrices. This paper is concerned with regularization of nonsymmetric stiffness matrices that arise from the boundary element method (BEM) for linear elasticity. Existence and uniqueness issues, as well as properties of the displacement field, for elasticity problems with tractions prescribed at every point on the boundary, are discussed in this paper.

Sufficient Conditions for Regularity and Singularity of Interval Matrices

Several verifiable sufficient conditions for regularity and singularity of interval matrices are given. As an application, a verifiable sufficient condition is derived for an interval matrix to have all eigenvalues real.

Theorems of the Alternative for Cones and Lyapunov Regularity of Matrices

Standard facts about separating linear functionals will be used to determine how two cones C and D and their duals C* and D* may overlap. When T: V → W is linear and K ⊂ V and D ⊂ W are cones, these results will be applied to C = T(K) and D, giving a unified treatment of several theorems of the alternate which explain when C contains an interior point of D. The case when V = W is the space H of n × n Hermitian matrices, D is the n × n positive semidefinite matrices, and T(X) = AX + X* A yields new and known results about the existence of block diagonal X's satisfying the Lyapunov condition: T(X) is an interior point of D. For the same V, W and D, T(X) = X − B* XB will be studied for certain cones K of entry-wise nonnegative X's.

Strong regularity of matrices in a discrete bounded bottleneck algebra

The results concerning strong regularity of matrices over bottleneck algebras are reviewed. We extend the known conditions to the discrete bounded case and modify the known algorithms for testing strong regularity.

Unique solvability of max-min fuzzy equations and strong regularity of matrices over fuzzy algebra

A matrix is said to have strongly linearly independent columns (or, in the case of square matrices, to be strongly regular) if for some vector b the system A ⊗ x = b has a unique solution. We formulate a necessary and sufficient condition for a linear system of equations over a fuzzy algebra to have a unique solution and prove the equivalence of strong regularity and trapezoidal property. Moreover, an algorithm for testing these properties is reviewed.

A note on checking regularity of interval matrices

It is proved that two previously known sufficient conditions for regularity of interval matrices are equivalent in the sense that they cover the same class of interval matrices.