A NEW PERTURBATION THEOREM FOR MOORE-PENROSE METRIC GENERALIZED INVERSE OF BOUNDED LINEAR OPERATORS IN BANACH SPACES

In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂ which is quite different from the method in [12] where ＂the generalized Banach lemma＂ was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.

A note on the Moore-Penrose inverse of a companion matrix

Let R be an associative ring with unity 1. The existence of the Moore-Penrose inverses of block matrices overR is investigated and the sufficient ad necessary conditions for such existence are obtained. Furthermore, the representation of the Moore-Penrose inverse of M=[0 A C B]is given under the condition of EBF - 0, where E - I - CCT and F - I -AfA. This result generalizes the representation of the Moore-Penrose inverse of the companion matrix M =[0 a In b]due to Pedro Patricio. As for applications, some examples are given to illustrate the obtained results.

The Reverse Order Law for Moore-Penrose Inverse of Closed Operators

In this paper, the reverse order law for the Moore-Penrose inverse of closed linear operators with closed range is investigated by virtue of the Norm-preserving extension of the bounded linear operators. The results generalize some results obtained by S Izumino in [12].

一个2×2块矩阵的Moore-Penrose逆的显式公式

The representation for the Moore-Penrose inverse of the matrix[AC BD]is derived by using the solvability theory of linear equations,where A∈C^(m×n),B∈C^(m×p),C∈C^(q×n)and D∈C^(q×p),with which some special cases are discussed.

Moore-Penrose Inverses and Group Inverses of Block k-Circulant Matrices

The Moore-Penrose inverse of a block k-circulant matrix whose blocks are arbitrary matrices are obtained when k has unit modulus. In the meantime. explicit formulae for finding group inverses of certain specified k-circulant matrices are also given.

Unified Approach to the Expression for the Moore-Penrose Inverse of a- xy

Let A be an unital C*-algebra, a, x and y are elements in A. In this paper, we present a method how to calculate the Moore-Penrose inverse of a- xy*and investigate the expression for some new special cases of(a- xy*).

Computing the Moore-Penrose Inverse of a Matrix Through Symmetric Rank-One Updates

This paper presents a recursive procedure to compute the Moore-Penrose inverse of a matrix A. The method is based on the expression for the Moore-Penrose inverse of rank-one modified matrix. The computational complexity of the method is analyzed and a numerical example is included. A variant of the algorithm with lower computational complexity is also proposed. Both algorithms are tested on randomly generated matrices. Numerical performance confirms our theoretic results.

Position Control of Flexible Joint Carts Using Adaptive Generalized Dynamics Inversion

This paper presents the design and implementation of Adaptive Generalized Dynamic Inversion(AGDI)to track the position of a Linear Flexible Joint Cart(LFJC)system along with vibration suppression of the flexible joint.The proposed AGDI control law will be comprised of two control elements.The baseline(continuous)control law is based on principle of conventional GDI approach and is established by prescribing the constraint dynamics of controlled state variables that reflect the control objectives.The control law is realized by inverting the prescribed dynamics using dynamically scaledMoore-Penrose generalized inversion.To boost the robust attributes against system nonlinearities,parametric uncertainties and external perturbations,a discontinuous control law will be augmented which is based on the concept of sliding mode principle.In discontinuous control law,the sliding mode gain is made adaptive in order to achieve improved tracking performance and chattering reduction.The closed-loop stability of resultant control law is established by introducing a positive define Lyapunov candidate function such that semi-global asymptotic attitude tracking of LFJC system is guaranteed.Rigorous computer simulations followed by experimental investigation will be performed on Quanser’s LFJC system to authenticate the feasibility of proposed control approach for its application to real world problems.

Perturbation Analysis for t-Product-Based Tensor Inverse,Moore-Penrose Inverse and Tensor System

This paper establishes some perturbation analysis for the tensor inverse,the tensor Moore-Penrose inverse,and the tensor system based on the t-product.In the settings of structured perturbations,we generalize the Sherman-Morrison-Woodbury(SMW)formula to the t-product tensor scenarios.The SMW formula can be used to perform the sensitivity analy-sis for a multilinear system of equations.

Moore-Penrose Inverse and Semilinear Equations

In this paper, we study the existence of solutions for the semilinear equation , where A is a , , and is a nonlinear continuous function. Assuming that the Moore-Penrose inverse AT(AAT)-1?exists (A denotes the transposed matrix of A) which is true whenever the determinant of the matrix AAT is different than zero, and the following condition on the nonlinear term satisfied . We prove that the semilinear equation has solutions for all. Moreover, these solutions can be found from the following fixed point relation .

A Novel Iterative Method to Find the Moore-Penrose Inverse of a Tensor with Einstein Product

In this study,based on an iterative method to solve nonlinear equations,a third-order convergent iterative method to compute the Moore-Penrose inverse of a tensor with the Einstein product is presented and analyzed.Numerical compar-isons of the proposed method with other methods show that the average number of iterations,number of the Einstein products,and CPU time of our method are considerably less than other methods.In some applications,partial and fractional differential equations that lead to sparse matrices are considered as prototypes.We use the iterates obtained by the method as a preconditioner,based on tensor form to solve the multilinear system A∗N X=B.Finally,several practical numerical examples are also given to display the accuracy and efficiency of the new method.The presented results show that the proposed method is very robust for obtaining the Moore-Penrose inverse of tensors.

Characterizations of EP, normal and Hermitian elements in rings using generalized inverses

The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R , then a is EP if and only if aa a^# = a^#aa . At the same time, the equivalent characterizations of a regular element to be EP are discussed.Let a∈R, then there exist b∈R such that a = aba and a is EP if and only if a∈R , a = a ba. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R , then a is normal if and only if a^＊a = a a^＊. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R ∩R^#, n∈N, then a is normal if and only if a^＊ a^＋（ a^#） n = a^# a＊（ a^＋） ^n. The results generalize the conclusions of Mosiet al.

CONTINUITY OF GENERALIZED INVERSES OF LINEAR OPERATORS IN BANACH SPACES AND ITS APPLICATIONS

The perturbation problem of generalized inverse is studied. And some new stability characteristics of generalized inverses were presented. It was also proved that the stability characteristics of generalized inverses were independent of the choice of the generalized inverse. Based on this result, two sufficient and necessary conditions for the lower semi-continuity of generalized inverses as the set-valued mappings are given.

Inverse displacement analysis of the general six degree-of-freedom serial robot based on optimization method

The paper presents a new solution of inverse displacement analysis of the general six degree-of-freedom serial robot.The inverse displacement analysis of the general serial robot is transformed into a minimization problem and then the optimization method is adopted to solve the nonlinear least squares problem with the analytic form of new Jacobian matrix.In this way,joint variables of the general serial robot can be searched out quickly under the desired precision when positions of the three non-collinear end effector points are given.Compared with the general Newton iterative method,the proposed algorithm can search out the solution when the robot is at the singular configuration and the initial configuration used in the optimization method may also be the singular configuration.So the convergence domain is bigger than that of the general Newton iterative method.Another advantage of the proposed algorithm is that positions of the three non-collinear end effector points are usually much easier to be measured than the orientation of the end effector.The inverse displacement analysis of the general 6R(six-revolute-joint) serial robot is illustrated as an example and the simulation results verify the efficiency of the proposed algorithm.Because the three non-collinear points can be selected at random,the method can be applied to any other types of serial robots.

Pressure-Controlled Inverse Ratio Ventilation during General Anesthesia for Open Abdominal Surgery Improves Postoperative Pulmonary Function

Background: Studies have shown that pressure-controlled ventilation improves alveolar gas distribution. And inverse ratio ventilation has advantages of improving oxygenation in acute respiratory distress syndrome (ARDS) patients. However, the effects that pressure-controlled inverse ration ventilation in patients undergoes endotracheal intubation general anesthesia have not been assessed. Objective: To investigate whether pressure-controlled inverse ratio ventilation (PIV) would improve ventilatory and oxygenation parameters as well as lung function compared to conventional ventilation in patients undergoing open abdominal surgery. Interventions: In the conventional ventilation (CV) group, the ventilation strategy involved zero end-expiratory pressure and volume-controlled ventilation. In the pressure-controlled inverse ratio ventilation (PIV) group, the strategy involved P high starting at 7 cm H2O, P low starting at 4 cm H2O, T high at 4 s, T low at 2 s, and an inspiratory-to-expiratory time ratio of 2:1. The ΔP value was adjusted according to VT. Pressure levels were increased by 2 cm H2O until a maximal VT was observed. Inspired oxygen fraction (FIO2) was 1.0 and tidal volume (VT) was 6 mL/kg in both groups. Main Outcome Measures: The primary outcome is pulmonary function tests. Hemodynamic, ventilatory and oxygenation parameters were measured;visual analog scale (VAS) scores, and nausea and vomiting scores were also measured. Results: The PIV group tolerated pressure-controlled inverse ratio ventilation without significant hemodynamic instability. Mean airway pressure and static compliance were significantly higher in the PIV group than those in CV group (P P 2 h was well tolerated and improved respiratory compliance and lung function on the first postoperative day.

THE METRIC GENERALIZED INVERSE AND ITS SINGLE-VALUE SELECTION IN THE PRICING OF CONTINGENT CLAIMS IN AN INCOMPLETE FINANCIAL MARKET

This article continues to study the research suggestions in depth made by M.Z.Nashed and G.F.Votruba in the journal"Bull.Amer.Math.Soc."in 1974.Concerned with the pricing of non-reachable"contingent claims"in an incomplete financial market,when constructing a specific bounded linear operator A:l_(1)^(n)→l_(2) from a non-reflexive Banach space l_(1)^(n) to a Hilbert space l_(2),the problem of non-reachable"contingent claims"pricing is reduced to researching the(single-valued)selection of the(set-valued)metric generalized inverse A■ of the operator A.In this paper,by using the Banach space structure theory and the generalized inverse method of operators,we obtain a bounded linear single-valued selection A^(σ)=A+of A■.

Two Families of Multipoint Root-Solvers Using Inverse Interpolation with Memory

In this paper, a general family of derivative-free n + 1-point iterative methods using n + 1 evaluations of the function and a general family of n-point iterative methods using n evaluations of the function and only one evaluation of its derivative are constructed by the inverse interpolation with the memory on the previous step for solving the simple root of a nonlinear equation. The order and order of convergence of them are proved respectively. Finally, the proposed methods and the basins of attraction are demonstrated by the numerical examples.

New Identities for Moore-Penrose Inverses of Some Operator Products and Their Reverse-order Laws

We establish new identities for Moore-Penrose inverses of some operator products,and prove their associated reverse-order laws.Moreover,our results concerning the Moore-Penrose inverse of a product of two operators lead in finding a relation between the operators in the case where Greville's inclusions are made into equalities.

L^p APPROXIMATION BY GENERAL BERNSTEIN-DURRMEYER OPERATOR DEFINED ON SIMPLEX

As a generalization of the Bernstein-Durrmeyer operatora defined on the simplex, a class of general Bernstein-Durrmeyer operators is introduced. With the weighted moduli of smoothness as a metric, we prove a strong direct theorem and an inverse theorem of weak type for these operators by using a decom-position way. From the theorems the characterization of Lp approximation behavior is derived.

ON THE GENERALIZED RESOLVENT OF LINEAR PENCILS IN BANACH SPACES

. Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvent of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil λ→ T - λS are provided and an explicit expression of the generalized resolvent is also given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area.