The Monotonicity Problems for Generalized Inverses of Matrices in H (n, ≥)

On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial order,give the explicit formulations of the monotonicity solution sets A{1;≥,T_1;≤B^(1)}and B{1;≥,T_2≥A^(1)}for the(1)-inverse,and two results of the monotonicity charac teriaztion for the(1,2)-inverse.

A Coordinate-Free Approach to the Design of Generalized Griffis-Duffy Platforms

Architectural singularity belongs to the Type II singularity,in which a parallel manipulator(PM)gains one or more degrees of freedom and becomes uncontrollable.PMs remaining permanently in a singularity are beneficial for linearto-rotary motion conversion.Griffis-Duffy(GD)platform is a mobile structure admitting a Bricard motion.In this paper,we present a coordinate-free approach to the design of generalized GD platforms,which consists in determining the shape and attachment of both the moving platform and the fixed base.The generalized GD platform is treated as a combination of six coaxial single-loop mechanisms under the same constraints.Owing to the inversion,hidden in the geometric structure of these single-loop mechanisms,the mapping from a line to a circle establishes the geometric transformation between the fixed base and the moving platform based on the center of inversion,and describes the shape and attachment of the generalized GD platform.Moreover,the center of inversion not only identifies the location of rotation axis,but also affects the shape of the platform mechanism.A graphical construction of generalized GD platforms using inversion,proposed in the paper,provides geometrically feasible solutions of the manipulator design for the requirement of the location of rotation axis.

The SMW Formula for Bounded Homogeneous Generalized Inverses with Applications

In this paper, we present an extension of the so-called classical Sherman-Morrison-Woodbury (for short SMW) formula for bounded homogeneous generalized inverse in Banach spaces. Some particular cases and applications will be also considered. Our results generalize the results of many authors for finite dimensional matrices and Hilbert space operators in the literature.

Some additive results for the generalized Drazin inverse in a Banach algebra

Let a, b be two generalized Drazin invertible elements in a Banach algebra. An explicit expression for the generalized Drazin inverse of the sum a + b in terms of a,b,a^d,b^d is given. The generalized Drazin inverse for the sum of two elements in a Banach algebra is studied by means of the system of idempotents. It is first proved that a + b∈A^(qnil) under the condition that a,b∈A^(qnil),aba = 0 and ab^2= 0 and then the explicit expressions for the generalized Drazin inverse of the sum a + b under some newconditions are given. Also, some known results are extended.

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.

PROPER SPLITTINGS FOR RESTRICTED LINEAR EQUATIONS AND THE GENERALIZED INVERSE A_(T,S)~2

This paper presents a proper splitting iterative method for comparing the general restricted linear euqations Ax=b, x ∈T （where, b ∈AT, and T is an arbitrary but fixed subspace of Cm） and the generalized in AT,S For the special case when b ∈AT and dim（T）=dim（AT）, this splitting iterative methverse AT,S hod converges to AT,Sb （the unique solution of the general restricted system Ax=bx ∈T）.

Entropy Bayesian Analysis for the Generalized Inverse Exponential Distribution Based on URRSS

This paper deals with the Bayesian estimation of Shannon entropy for the generalized inverse exponential distribution.Assuming that the observed samples are taken from the upper record ranked set sampling(URRSS)and upper record values(URV)schemes.Formulas of Bayesian estimators are derived depending on a gamma prior distribution considering the squared error,linear exponential and precautionary loss functions,in addition,we obtain Bayesian credible intervals.The random-walk Metropolis-Hastings algorithm is handled to generate Markov chain Monte Carlo samples from the posterior distribution.Then,the behavior of the estimates is examined at various record values.The output of the study shows that the entropy Bayesian estimates under URRSS are more convenient than the other estimates under URV in the majority of the situations.Also,the entropy Bayesian estimates perform well as the number of records increases.The obtained results validate the usefulness and efficiency of the URV method.Real data is analyzed for more clarifying purposes which validate the theoretical results.

GENERALIZED WAVELETS AND INVERSION OF THE RADON TRANSFORM ON THE LAGUERRE HYPERGROUP

Let X=Rn +×R denote the underlying manifold of polyradial functions on the Heisenberg group H n. We construct a generalized translation on X=Rn +×R, and establish the Plancherel formula on L2(X,dμ). Using the Gelfand transform we give the condition of generalized wavelets on L2(X,dμ). Moreover, we show the reconstruction formulas for wavelet packet trnasforms and an inversion formula of the Radon transform on X.

A NOTE ON THE REPRESENTATIONS FOR THE GENERALIZED DRAZIN INVERSE OF BLOCK MATRICES

We present some representations for the generalized Drazin inverse of a block matrix x =[cd ab]in a Banach algebra ~4 in terms of ad and （be）d under certain conditions,extending some recent result related to the generalized Drazin inverse of an anti-triangular operator matrix. Also, several particular cases of this result are considered.

Duplicate Form of the Generalized Carlitz Inversions and Summation Formulae

The duplicate form of the generalized Gould-Hsu inversions has been obtained by Shi and Zhang. In this paper, we present a simple proof of this duplicate form. With the same method, we construct the duplicate form of the generalized Carlitz inversions. Using this duplicate form, we obtain several terminating basic hypergeometric identities and some limiting cases.

A CHARACTERIZATION OF THE GENERALIZED INVERSES OVER DIVISION RINGS

Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and necessary condition on the existence ofthe Moore-Penrose inverse over any division rings.Our result can be regarded as an im-provement of Theorem lin[1].As a medium result,we also show a characterization ofthe{1,2}-inverse.

The Discriminance for <i>FLDcirc_r</i>Matrices and the Fast Algorithm of Their Inverse and Generalized Inverse

This paper presents a new type of circulant matrices. We call it the first and the last difference r-circulant matrix (FLDcircr matrix). We can verify that the linear operation, the matrix product and the inverse matrix of this type of matrices are still FLDcircr matrices. By constructing the basic FLDcircr matrix, we give the discriminance for FLDcircr matrices and the fast algorithm of the inverse and generalized inverse of the FLDcircr matrices.

Site effects by generalized inversion technique using strong motion recordings of the 2008 Wenchuan earthquake

The generalized inversion of S-wave amplitude spectra from the free-field strong motion recordings of the China National Strong Motion Observation Network System （NSMONS） are used to evaluate the site effects in the Wenchuan area. In this regard, a total of 602 recordings from 96 aftershocks of the Wenchuan earthquake with magnitudes of M3.7-M6.5 were selected as a dataset. These recordings were obtained from 28 stations at a hypocenter distance ranging from 30 km to 150 km. The inversion results have been verified as reliable by comparing the site response at station 62WUD using the Generalized Inversion Technique （GIT） and the Standard Spectral Ratio method （SSR）. For all 28 stations, the site predominant frequency F and the average site amplification in different frequency bands of 1.0-5.0 Hz, 5.0-10.0 Hz and 1.0-10.0 Hz have been calculated based on the inversion results. Compared with the results from the horizontal-to-vertical spectral ratio （HVSR） method, it shows that the HVSR method can reasonably estimate the site predominant frequency but underestimates the site amplification. The linear fitting between the average site amplification for each frequency band and the V20 （the average uppermost-20 m shear wave velocity） shows good correlation. A distance measurement called the asperity distance DAspt is proposed to reasonably characterize the source-to-site distance for large earthquakes. Finally, the inversed site response is used to identify the soil nonlinearity in the main shock and aftershocks of Wenchuan earthquake. In ten of the 28 stations analyzed in the main shock, the soil behaved nonlinearly, where the ground motion level is apparently beyond a threshold ofPGA 〉 300 cm/s^2 or PGV 〉 20 cm/s, and only one station coded 51SFB has evidence of soil nonlinear behavior in the aftershocks.

SPECTRUM DISTRIBUTION OF THE SEDOND ORDER GENERALIZED DISTRIBUTED PARAMETER SYSTEMS

Spectrum distribution of the second order generalized distributed parameter system was discussed via the functional analysis and operator theory in Hilbert space. The solutions of the problem and the constructive expression of the solutions are given by the generalized inverse one of bounded linear operator. This is theoretically important for studying the stabilization and asymptotic stability of the second order generalized distributed parameter system.

An Efficient Tide-Surge Interaction Model for the Coast of Bangladesh

The numerical method of lines(MOLs)in coordination with the classical fourth-order Runge Kutta(RK(4,4))method is used to solve shallow water equations(SWEs)for foreseeing water levels owing to the nonlinear interaction of tide and surge accompanying with a storm along the coast of Bangladesh.The SWEs are developed by extending the body forces with tide generating forces(TGFs).Spatial variables of the SWEs along with the boundary conditions are approximated by means of finite difference technique on an Arakawa C-grid to attain a system of ordinary differential equations(ODEs)of initial valued in time,which are being solved with the aid of the RK(4,4)method.Nested grid technique is adopted to solve coastal complexities closely with least computational cost.A stable tidal solution in the region of our choice is produced by applying the tidal forcing with the major tidal constituent M2(lunar semi-diurnal)along the southern open-sea boundary of the outer scheme.Numerical experimentations are carried out to simulate water levels generated by the cyclonic storm AILA along the coast of Bangladesh.The model simulated results are found to be in a reasonable agreement with the limited available reported data and observations.

Generalized Inverse Analysis on the Domain ?(A, A^+) in B(E,F)

Let B(E,F) be the set of all bounded linear operators from a Banach space E into another Banach space F,B^+(E, F) the set of all double splitting operators in B(E, F)and GI(A) the set of generalized inverses of A ∈ B^+(E, F). In this paper we introduce an unbounded domain ?(A, A^+) in B(E, F) for A ∈ B^+(E, F) and A^+∈GI(A), and provide a necessary and sufficient condition for T ∈ ?(A, A^+). Then several conditions equivalent to the following property are proved: B = A+(IF+(T-A)A^+)^(-1) is the generalized inverse of T with R(B)=R(A^+) and N(B)=N(A^+), for T∈?(A, A^+), where IF is the identity on F. Also we obtain the smooth(C~∞) diffeomorphism M_A(A^+,T) from ?(A,A^+) onto itself with the fixed point A. Let S = {T ∈ ?(A, A^+) : R(T)∩ N(A^+) ={0}}, M(X) = {T ∈ B(E,F) : TN(X) ? R(X)} for X ∈ B(E,F)}, and F = {M(X) : ?X ∈B(E, F)}. Using the diffeomorphism M_A(A^+,T) we prove the following theorem: S is a smooth submanifold in B(E,F) and tangent to M(X) at any X ∈ S. The theorem expands the smooth integrability of F at A from a local neighborhoold at A to the global unbounded domain ?(A, A^+). It seems to be useful for developing global analysis and geomatrical method in differential equations.

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*).

The Generalized Inverse of Singular Integral Operators on an Open Arc and Its Applications

After choosing weight functions suitably, we define a Banach spaceH ω μ (L) and discuss the generalized inverse of singular integral operators on an open arc. Using the generalized inverse, we obtain the solutions for the following singular integral equation

The Explicit Expressions of Level- k(r1 ,r2 ,… ,rk) Circulant Matrices and Their Some Properties

In this paper,we give the explicit expressions of level k (r 1,r 2,…,r k) circulant matrices of order n 1n 2…n k,and the explicit expressions for the eigenvalues,the determinants and the inverse matrices of the kind level k (r 1,r 2,…,r k) circulant matrices are derived,and it is also proved that the sort of matrices are diagonalizable.

The Explicit Expressions of Level-k Circulant Matrices of Type(n_1,n_2.…,n_k) and Their Some Properties

In this paper, we give the explicit expressions of level-k circulant matrices of type (n1,n2,…nk) and of order n1n2…nk,and the explicit expressions for the eigenvalues,the determinants and the inverse matrices of the kind level-k circulant matrices are derived,and it is also proved that the sort matrices are unitarily diagonalizable.