ON NEAR-RINGS WITH MATRIX UNITS

Abstract

In ring theory it is well known that a ring R with identity is isomorphic to a matrix ring if and only if R has a set of matrix units. In this paper, the above result is extended to matrix near-rings and it is proved that a near-ring R with identity is isomorphic to a matrix near-ring if and only if R has a set of matrix units and satisfies two other conditions. As a consequence of this result several examples of matrix near-rings are given and for a finite group (Γ, +) with o(Γ) > 2 it is proved that M₀ (Γⁿ) is (isomorphic to) a matrix near-ring.     

Two-grid finite element method for the dual-permeability-Stokes fluid flow model

In this paper, two-grid finite element method for the steady dual-permeability-Stokes fluid flow model is proposed and analyzed. Dual-permeability-Stokes interface system has vast applications in many areas such as hydrocarbon recovery process, especially in hydraulically fractured tight/shale oil/gas reservoirs. Two-grid method is popular and convenient to solve a large multiphysics interface system by decoupling the coupled problem into several subproblems. Herein, the two-grid approach is used to reduce the coding task substantially, which provides computational flexibility without losing the approximate accuracy. Firstly, we solve a global problem through standard P_k ? P_k??1 ? P_k ? P_k finite elements on the coarse grid. After that, a coarse grid solution is applied for the decoupling between the interface terms and the mass exchange terms to solve three independent subproblems on the fine grid. The three independent parallel subproblems are the Stokes equations, the microfracture equations, and the matrix equations, respectively. Four numerical tests are presented to validate the numerical methods and illustrate the features of the dual-permeability-Stokes model.

     

On the Absolute Matrix Summability Factors of Fourier Series

In this paper, two known theorems on |N?, p _n | _k summability methods of Fourier series have been generalized for |A, p _n | _k summability factors of Fourier series by using different matrix transformations. New results have been obtained dealing with some other summability methods.

     

Transitioning Between Tableaux and Spider Bases for Specht Modules

Regarding the Specht modules associated to the two-row partition (n, n), we provide a combinatorial path model to study the transitioning matrix from the tableau basis to the A₁-web basis (i.e. cup diagrams), and prove that the entries in this matrix are positive in the upper-triangular portion with respect to a certain partial order.

     

Fast computations with the harmonic Poincaré–Steklov operators on nested refined meshes

In this paper we develop asymptotically optimal algorithms for fast computations with the discrete harmonic Poincaré–Steklov operators (Dirichlet–Neumann mapping) for interior and exterior problems in the presence of a nested mesh refinement. Our approach is based on the multilevel interface solver applied to the Schur complement reduction onto the nested refined interface associated with a nonmatching decomposition of a polygon by rectangular substructures. This paper extends methods from Khoromskij and Prössdorf (1995), where the finite element approximations of interior problems on quasi‐uniform grids have been considered. For both interior and exterior problems, the matrix–vector multiplication with the compressed Schur complement matrix on the interface is shown to have a complexity of the order O(N _r log³ N _u), where N_r = O((1 + p _r) N _u) is the number of degrees of freedom on the polygonal boundary under consideration, N _u is the boundary dimension of a finest quasi‐uniform level and p _r ⩾ 0 defines the refinement depth. The corresponding memory needs are estimated by O(N _r log^q N _u), where q = 2 or q = 3 in the case of interior and exterior problems, respectively.

     

Computing the nearest stable matrix pairs

In this paper, we study the nearest stable matrix pair problem: given a square matrix pair (E,A), minimize the Frobenius norm of (Δ_E,Δ_A) such that (E+Δ_E,A+Δ_A) is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian matrix pairs: A matrix pair (E,A) is dissipative Hamiltonian if A=(J−R)Q with skew‐symmetric J, positive semidefinite R, and an invertible Q such that Q^TE is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.     

Numerical investigation into the dependence of the Allen–Cahn equation on the free energy

In this paper, we study a direct parallel-in-time (PinT) algorithm for first- and second-order time-dependent differential equations. We use a second-order boundary value method as the time integrator. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix B, which yields a direct parallel implementation across all time steps. A crucial issue of this methodology is how the condition number (denoted by Cond₂(V )) of the eigenvector matrix V of B behaves as n grows, where n is the number of time steps. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for V and V^− 1, by which we prove that Cond\(_{2}(V)=\mathcal {O}(n^{2})\). This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows, and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. A fast structure-exploiting algorithm is also designed for computing the spectral diagonalization of B. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.

     

Free and Non-Free Subgroups of the Group UT(∞, ℤ)

We provide a method to find free groups of rank two in the group of infinite unitriangular matrices. Our groups are generated by two block-diagonal matrices, namely of the form A = diag(C, C, C…), B = diag(I _t , C, C…), where C is a matrix of finite dimension.

We give a necessary and sufficient condition for A and B defined above to generate a free group when C is a transvection. We formulate a sufficient condition to generate a free group, when C is a product of any number of commuting transvections.

We provide a classification of groups defined above, when C is of degree 3 or 4.     

Methode numerique de detection de la singularite d'une matrice

The detection of matrix singularity by a numerical method is not easily accomplished on account of the finite precision of computer arithmetic.In this paper we present a numerical method which determines a valueC for the degree of conditioning of a matrix. This value isC=0 for a singular matrix and has progressively larger values for matrices which are increasingly well-conditioned. This value isC=C _max (C _max defined by the precision of the computer) when the matrix is perfectly well-conditioned.     

The quaternion matrix equation ΣA i XB i =E

LetH _F be the generalized quaternion division algebra over a fieldF with charF#2. In this paper, the adjoint matrix of anyn×n matrix overH _F [γ] is defined and its properties is discussed. By using the adjoint matrix and the method of representation matrix, this paper obtains several necessary and sufficient conditions for the existence of a solution or a unique solution to the matrix equation Σ _i=0 ^k A ⁱ XB _i =E overH _F , and gives some explicit formulas of solutions. Supported by the National Natural Science Foundation of China and Human     

An efficient algorithm for the generalized centro-symmetric solution of matrix equation <Emphasis Type="Italic">A X B</Emphasis> = <Emphasis Type="Italic">C</Emphasis>

In this paper, an iterative algorithm is constructed for solving linear matrix equation AXB = C over generalized centro-symmetric matrix X. We show that, by this algorithm, a solution or the least-norm solution of the matrix equation AXB = C can be obtained within finite iteration steps in the absence of roundoff errors; we also obtain the optimal approximation solution to a given matrix X ₀ in the solution set of which. In addition, given numerical examples show that the iterative method is efficient.     

Extensions of Rings with Many Units

Abstract

In this paper, we investigate stable range conditions over extensions of matrix rings. It is shown that a ring R satisfies (s, 2)-stable range if and only if R has a complete orthogonal set {e ₁,…, e _n } of idempotents such that all e _i Re _i satisfy (s, 2)-stable range. Also we extend this result to (s, 2)-rings and rings satisfying unit 1-stable range.     

Generalization of Strang's Preconditioner with Applications to Toeplitz Least Squares Problems

In this paper, we propose a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices A_n. The th column of our circulant preconditioner Sn is equal to the th column of the given matrix A_n. Thus if A_n is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be defined as . This construction is similar to the forward-backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. We show that if the matrix A_n has decaying coefficients away from the main diagonal, then is a good preconditioner for An. Comparisons of our preconditioner with other circulant-based preconditioners are carried out for some 1-D Toeplitz least squares problems: min ∥ b - Ax∥₂. Preliminary numerical results show that our preconditioner performs quite well, in comparison to other circulant preconditioners. Promising test results are also reported for a 2-D deconvolution problem arising in ground-based atmospheric imaging.     

An efficient iterative method for solving the matrix equation AXB + CYD = E

This paper presents an iterative method for solving the matrix equation AXB + CYD = E with real matrices X and Y. By this iterative method, the solvability of the matrix equation can be determined automatically. And when the matrix equation is consistent, then, for any initial matrix pair [X₀, Y₀], a solution pair can be obtained within finite iteration steps in the absence of round‐off errors, and the least norm solution pair can be obtained by choosing a special kind of initial matrix pair. Furthermore, the optimal approximation solution pair to a given matrix pair [X?, ?] in a Frobenius norm can be obtained by finding the least norm solution pair of a new matrix equation AX?B + C?D = ?, where ? = E ? AX?B ? C?D. The given numerical examples show that the iterative method is efficient. Copyright © 2005 John Wiley & Sons, Ltd.     

GOOD IDEALS IN MATRIX RINGS OVER COMMUTATIVE PIR's

Abstract

An ideal A of a ring R is called a good ideal if the coset product r ₁ r ₂ + A of any two cosets r ₁ + A and r ₂ + A of A in the factor ring R/A equals their set product (r ₁ + A) º (r ₂ + A): = {(r ₁ + a)(r ₂ + a ₂): a ₁, a ₂ ε A}. Good ideals were introduced in [3] to give a characterization of regular right duo rings. We characterize the good ideals of blocked triangular matrix rings over commutative principal ideal rings and show that the condition A º A = A is sufficient for A to be a good ideal in this class of matrix rings, none of which are right duo. It is not known whether good ideals in a base ring carries over to good ideals in complete matrix rings over the base ring. Our characterization shows that this phenomenon occurs indeed for complete matrix rings of certain sizes if the base ring is a blocked triangular matrix ring over a commutative principal ideal ring.     

Operational Matrix of Fractional Integration Based on the Shifted Second Kind Chebyshev Polynomials for Solving Fractional Differential Equations

This paper is devoted to studying a computational method for solving multi-term differential equations based on new operational matrix of shifted second kind Chebyshev polynomials. The properties of the operational matrix of fractional integration are exploited to reduce the main problem to an algebraic equation. We present an upper bound for the error in our estimation that leads to achieve the convergence rate of O(M ^−κ). Numerical experiments are reported to demonstrate the applicability and efficiency of the proposed method.

     

Some intriguing upper bounds for separating hash families

An N ×n matrix on q symbols is called {w_1,...,w_t}-separating if for arbitrary t pairwise disjoint column sets C_1,..., C_t with |C_i|=w_i for 1 ≤i≤t, there exists a row f such that f(C_1),...,f(C_t) are also pairwise disjoint, where f(C_i) denotes the collection of componentn of C_i restricted to row f. Given integers N, q and w_1,...,w_t, denote by C(N,q,{w_1,...,w_t}) the maximal a such that a corresponding matrix does exist.The determination of C(N,q,{w_1,...,w_t}) has received remarkable attention during the recent years. The main purpose of this paper is to introduce two novel methodologies to attack the upper bound of C(N, q, {w_1,...,w_t}).The first one is a combination of the famous graph removal lemma in extremal graph theory and a Johnson-type recursive inequality in coding theory, and the second onc is the probabilistic method. As a consequence, we obtain several intriguing upper bounds for some parameters of C(N,q,{w_1,...,w_t}), which significantly improve the previously known results.     

Some extensions of the improved modified Gauss–Seidel iterative method for H‐matrices

In this paper, first we present a convergence theorem of the improved modified Gauss–Seidel iterative method, referred to as the IMGS method, for H‐matrices and compare the range of parameters α_i with that of the parameter ω of the SOR iterative method. Then with a more general splitting, the convergence analysis of this method for an H‐matrix and its comparison matrix is given. The spectral radii of them are also compared. Copyright © 2006 John Wiley & Sons, Ltd.     

Zeroth order H ∞ norm approximation of multivariable systems

In this paper the H _∞ norm approximation of a given stable, proper, rational transfer function by a constant matrix is considered (Zeroth order H _∞ norm approximation problem). The solution method is based on the observation that the H _∞ norm approximation problem can be put into an allpass imbedding problem.     

矩阵方程A1Z+ZB1=C1的广义自反最佳逼近解的迭代算法

本文研究了Sylvester复矩阵方程A_1Z+ZB_1=c_1的广义自反最佳逼近解.利用复合最速下降法,提出了一种的迭代算法.不论矩阵方程A_1Z+ZB_1=C_1是否相容,对于任给初始广义自反矩阵Z_0,该算法都可以计算出其广义自反的最佳逼近解.最后,通过两个数值例子,验证了该算法的可行性.