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In this paper, the concept of "Inverse Complemented Matrix Method", introduced by Eagambaram (2018), has been reestablished with the help of minus partial order and several new properties of complementary matrices and the inverse of complemented matrix are discovered. Class of generalized inverses and outer inverses of given matrix are characterized by identifying appropriate inverse complement. Further, in continuation, we provide a condition equivalent to the regularity condition for a matrix to have unique shorted matrix in terms of inverse complemented matrix. Also, an expression for shorted matrix in terms of inverse complemented matrix is given. [ABSTRACT FROM AUTHOR]

One of the fundamental research problems in the theory of generalized inverses has certainly been establishments of various matrix equalities that involve generalized inverses. A simplest form of such matrix equalities is given by A (i , ... , j) = B (k , ... , l) , where A and B are matrices of the same size, and (⋅) (i , ... , j) denotes the { i , ... , j } -generalized inverse of a matrix. Because generalized inverses of a matrix are not necessarily unique, the equality A (i , ... , j) = B (k , ... , l) does not imply A=B for singular matrices. In this paper, we derive necessary and sufficient conditions for this kind of equalities to hold for the eight commonly-used types of generalized inverse of A and B using the matrix equation method and the matrix rank method. [ABSTRACT FROM AUTHOR]

Fundamental in matrix algebra and its applications, a generalized inverse of a real matrix A is a matrix H that satisfies the Moore--Penrose (M--P) property AH A = A. If H also satisfies the additional useful M--P property, H AH = H, it is called a reflexive generalized inverse. Reflexivity is equivalent to minimum rank, so we are particularly interested in reflexive generalized inverses. We consider aspects of symmetry related to the calculation of a sparse reflexive generalized inverse of A. As is common, and following Fampa and Lee [Oper. Res. Lett., 46 (2018), pp. 605-610] for calculating sparse generalized inverses, we use (vector) 1-norm minimization for inducing sparsity and for keeping the magnitude of entries under control. When A is symmetric, we may naturally desire a symmetric H, while generally such a restriction on H may not lead to a 1-norm minimizing reflexive generalized inverse. We investigate a block construction method to produce a symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We provide a theoretically efficient and practical local-search algorithm to block construct an approximate 1-norm minimizing symmetric reflexive generalized inverse. Another aspect of symmetry that we consider relates to another M--P property: H is ah-symmetric if AH is symmetric. The ah-symmetry property is the key one for solving least-squares problems using H. Here we do not assume that A is symmetric, and we do not impose symmetry on H. We investigate a column block construction method to produce an ah-symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We provide a theoretically efficient and practical local-search algorithm to column block construct an approximate 1-norm minimizing ah-symmetric reflexive generalized inverse. [ABSTRACT FROM AUTHOR]

In this paper, we present equivalent characterizations for the minimal ∞-norm of generalized inverses of the incidence matrix of a tree. We also compute the minimal ∞-norm for a class of trees. [ABSTRACT FROM AUTHOR]

It is known that within the set of orthogonal projectors (Hermitian idempotent matrices) certain matrix partial orderings coincide in the sense that when two orthogonal projectors are ordered with respect to one of the orderings, then they are also ordered with respect to the others. This concerns, inter alia, the star, minus, diamond, sharp, core, and Löwner orderings. The situation changes, though, when instead of two orthogonal projectors, various functions of the pair (being either no longer Hermitian or no longer idempotent) are compared. The present paper provides an extensive investigation of the matrix partial orderings of functions of two orthogonal projectors. In addition to the six orderings mentioned above, three further binary functions are covered by the analysis, one of which is the space preordering. A particular attention is paid to the requirements that either product, sum, or difference of two orthogonal projectors is itself an orthogonal projector, i.e., inherits both features, Hermitianness and idempotency. Links of the results obtained with the research areas of applied origin (e.g., physics and statistics) are pointed out as well. [ABSTRACT FROM AUTHOR]

Generalized inverses are extremely effective in many areas of mathematics and engineering. The zeroing neural network (ZNN) technique, which is currently recognized as the state-of-the-art approach for calculating the time-varying Moore-Penrose matrix inverse, is investigated in this study as a solution to the problem of calculating the time-varying minimum rank outer inverse (TV-MROI) with prescribed range and/or TV-MROI with prescribed kernel. As a result, four novel ZNN models are introduced for computing the TV-MROI, and their efficiency is examined. Numerical tests examine and validate the effectiveness of the introduced ZNN models for calculating TV-MROI with prescribed range and/or prescribed kernel. • The TV-MROI with PR and/or with PK problems are addressed for the first time through the ZNN approach. • With the purpose of computing the TV-MROI with PR and/or PK, four novel ZNN models are introduced. • A theoretical investigation is performed on the models to validate them. [ABSTRACT FROM AUTHOR]

When sample survey data with complex design (stratification, clustering, unequal selection or inclusion probabilities, and weighting) are used for linear models, estimation of model parameters and their covariance matrices becomes complicated. Standard fitting techniques for sample surveys either model conditional on survey design variables, or use only design weights based on inclusion probabilities essentially assuming zero error covariance between all pairs of population elements. Design properties that link two units are not used. However, if population error structure is correlated, an unbiased estimate of the linear model error covariance matrix for the sample is needed for efficient parameter estimation. By making simultaneous use of sampling structure and design-unbiased estimates of the population error covariance matrix, the paper develops best linear unbiased estimation (BLUE) type extensions to standard design-based and joint design and model based estimation methods for linear models. The analysis covers both with and without replacement sample designs. It recognises that estimation for with replacement designs requires generalized inverses when any unit is selected more than once. This and the use of Hadamard products to link sampling and population error covariance matrix properties are central topics of the paper. Model-based linear model parameter estimation is also discussed.

In this paper, firstly, we establish the inertia formulae for some matrix expressions related to the generalized inverse . Then, as applications, based on the derived inertia formulae, we study the definiteness of some matrices. Finally, necessary and sufficient conditions for some matrices to be positive definite, positive semidefinite, negative definite and negative semi-definite are given, respectively.

The quadrilateral graph Q(G) is obtained from G by replacing each edge in G with two parallel paths of length 1 and 3, whereas the pentagonal graph W(G) is obtained from G by replacing each edge in G with two parallel paths of length 1 and 4. In this paper, closed-form formulas of resistance distance and Kirchhoff index for quadrilateral graph and pentagonal graph are obtained whenever G is an arbitrary graph.

Let G be a connected graph. The subdivision graph S(G) of a graph (G) is the graph obtained by inserting a new vertex into every edge of G. The set of such new vertices is denoted by I(G). The generalized subdivision-vertex corona of G and Hi for i = 1, 2, ... , n, denoted by J(G) ⊙ Λi=1n Hi, is the graph obtained from S(G) and Hi by joining the ith vertex of V (G) to every vertex in Hi. The generalized subdivision-edge corona of G and Hi for i = 1, 2, ... , m, denoted by J(G) ⊖ Λi=1n Hi, is the graph obtained from S(G) and Hi by joining the ith vertex of I(G) to every vertex in Hi. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of J(G) ⊙ Λi=1n Hi and J(G) ⊖ Λi=1n Hi whenever G and Hi are an arbitrary graph. These results generalize the existing results.

Let b and c be two elements in a semigroup S. The (b,c)-inverse is an important outer inverse because it unifies many common generalized inverses. This paper is devoted to presenting some symmetric properties of (b,c)-inverses and (c,b)-inverses. We first find that S contains a (b,c)-invertible element if and only if it contains a (c,b)-invertible element. Then, for four given elements a,b,c,d in S, we prove that a is (b,c)-invertible and d is (c,b)-invertible if and only if abd is invertible along c and dca is invertible along b. Inspired by this result, the (b,c)-invertibility is characterized by one-sided invertible elements. Furthermore, we show that a is inner (b,c)-invertible and d is inner (c,b)-invertible if and only if c is inner (a,d)-invertible and b is inner (d,a)-invertible.

This paper considers how to construct and describe matrix equalities that are composed of algebraic operations of matrices and their generalized inverses. We select a group of known and new reverse-order laws for generalized inverses of several matrix products and derive various necessary and sufficient conditions for them to hold using the matrix rank method and the block matrix method.

The technology of damage evaluation based on data collected by sensors has become an important topic in the field of structural engineering This paper presents an improved generalized flexibility method for sensor placement and damage evaluation. Compared with the existing methods, the main improvements of the proposed method lie in three aspects. The first one is to replace mass-normalized mode shapes with arbitrary-scaled mode shapes since that only the latter can be obtained in modal testing under ambient excitation. The second one is to combine the generalized flexibility sensitivity with the frequency sensitivity to obtain more equations. The third one is to use a new regularized generalized inverse technique for computing the unknowns more accurately. Sensor placement based on the improved generalized flexibility method can be divided into three steps. Firstly, the number of sensors can be predicted by a simple formula derived by the principle of the number of equations should close to the number of unknowns as far as possible. Secondly, the key components of the whole structure are determined based on their contributions to the global generalized flexibility change. Thirdly, the sensor positions can be obtained according to the common nodes of those key components. Overall, the proposed method is simple and convenient since it only needs environment excitation and a few sensors. The results of numerical and experimental examples show that the proposed method can successfully evaluate structural damage with a few sensor signals. [ABSTRACT FROM AUTHOR]

Given elements a, b, c of any associative ring R with 1, then a is called right hybrid (b, c)-invertible if there exists y ∈ R such that yay = y , y R = b R and rann (y) = rann (c). It is shown that such y exists if and only if c ∈ cabR and rann (cab) ⊆ rann (b) , in which case y is unique. With an appropriate generalization of the right annhilator rann (.), this result is extended to all semigroups with 1 and to all categories. The semigroup version of right (and left) annihilator ideals is also used to define five finiteness properties for semigroups, and to establish the implications between them. Corresponding results for categories are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, novel representations of generalized inverses of rational matrices are developed. Therefore, a unified approach for the computation of {1,2,3} and {1,2,4} inverses and Moore-Penrose inverse of a given matrix A is considered. Full-rank QDR decomposition of a rational matrix is utilized to avoid the square roots of rational expressions in the evaluations, making the given algorithm very suitable for symbolic computations of generalized matrix inverses. Furthermore, we developed an algorithm for symbolic computation of the Moore-Penrose inverse of a polynomial matrix using the full-rank QDR decomposition, therefore maximizing the potential of using square root–free polynomial entries. Introduced algorithms are illustrated via numerical examples. [ABSTRACT FROM AUTHOR]

The dynamic properties of structural and mechanical systems are commonly predicted using frequency response functions (FRFs), which represent the relationship between input forces and the corresponding responses in the frequency domain. Analytically calculated responses seldom coincide with actual responses because of modeling and construction errors, performance deterioration in service, etc. The inconsistency requires the correction of the FRF matrix for subsequent analysis, performance evaluation, etc. Obtaining the complete FRF matrix corresponding to all the degrees of freedom at measurement is not practical. This paper derives the mathematical form to expand the incomplete FRF matrix at measurement using the least-squares approach constrained by measured FRFs and correct the FRF matrix at the intact state. Additionally, this paper provides two analytical methods to predict unknown input dynamic excitations using the expanded FRF matrix and direct approaches. Numerical applications elucidate the validity and drawbacks of the proposed methods. [ABSTRACT FROM AUTHOR]

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The generalized inverse has numerous important applications in aspects of the theoretic research of matrices and statistics. One of the core problems of generalized inverse is finding the necessary and sufficient conditions for the reverse (or the forward) order laws for the generalized inverse of matrix products. In this paper, by using the extremal ranks of the generalized Schur complement, some necessary and sufficient conditions are given for the forward order law for A1{1,2}A2{1,2}…An{1,2}⊆(A1A2…An){1,2}.

The current study investigates the solvability conditions and the general solution of three symmetrical systems of coupled Sylvester-like quaternion matrix equations. Accordingly, the necessary and sufficient conditions for the consistency of these systems are determined, and the general solutions of the systems are thereby deduced. An algorithm and a numerical example are constructed over the quaternions to validate the results of this paper.

The use of Rayleigh wave dispersion curve for estimating shear wave velocity (Vs) is a common task in the field of engineering geophysics. However, because of the nonlinear nature of Rayleigh wave dispersion curves, using a proper technique in the inversion procedure in order to find adequate model parameters is a challenging problem. In this study, a comparative study is performed between linear and nonlinear inversion methods. Between different techniques, a new nonlinear searching algorithm, i.e., the modified cuckoo search (MCS) algorithm was introduced to invert the Rayleigh wave dispersion curves. Also, the generalized inverse (GI) method was used as a linear searching algorithm applied to the construction of the Vs profile. The proposed algorithms were tested by different synthetic (noisy and noise free) datasets. The results show that the MCS algorithm has more exploration ability in comparison with the original cuckoo search. Then, an actual dataset was inverted by the introduced inversion methods. The results show that the performance of the MCS is better than that of the GI method. MCS is a fast, powerful and user-friendly technique in the field of dispersion curve inversion. [ABSTRACT FROM AUTHOR]

Matrix expressions composed by generalized inverses can generally be written as f(A1-, A2-,...., Ak-), where A1, A2,...., Ak are a family of given matrices of appropriate sizes, and (·)- denotes a generalized inverse of matrix. Once such an expression is given, people are primarily interested in its uniqueness (invariance property) with respect to the choice of the generalized inverses. As such an example, this article describes a general method for deriving necessary and sufficient conditions for the matrix equality A1A2- A3A4- A5 = A to always hold for all generalized inverses A2- and A4- of A2 and A4 through use of the block matrix representation method and the matrix rank method, and discusses some special cases of the equality for different choices of the five matrices. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and representations of the generalized inverse A T , ∗ (2) , A T , ∗ (1 , 2) , A ∗ , S (2) , A ∗ , S (1 , 2) , A T , S (2) , A T , S (1 , 2) of a matrix A over an associative ring, and give some characterizations and representations of these generalized inverses by using regularity. Some explicit expressions for A T , S (2) and A T , S (1 , 2) of a matrix A by means of { 1 } -inverse are also given. Furthermore, we present the relationship between the generalized inverse A T , S (2) and the (B , C) -inverse of a matrix A over rings. And then using the regularity give the sufficient and necessary condition for the existence and the expression of (B , C) -inverse of matrices over a ring. Some explicit expressions for (b , c) -inverse of an element in semigroups by means of { 1 } -inverse are also studied. As applications, some new results for the Mary inverse, the core inverse and dual core inverse are obtained. [ABSTRACT FROM AUTHOR]

Reverse order laws for generalized inverses of matrix products are a classic object of study in the theory of generalized inverses. One of the well-known reverse order laws for a matrix product AB is (AB)(i,…,j)=B(s2,…,t2)A(s1,…,t1), where (⋅)(i,…,j) denotes a {i,…,j}-generalized inverse of matrix. Because {i,…,j}-generalized inverse of a singular matrix is not unique, the relationships between both sides of the reverse order law can be divided into four situations for consideration. The aim of this paper is to give an overview of plenty of results concerning reverse order laws for {i,…,j}-generalized inverses of the product AB, from the development of background and preliminary tools to the collection of miscellaneous formulas and facts on the reverse order laws in one place with cogent introduction and references for further study. We begin with the introduction of a linear mixed model y=ABβ+Aγ+ϵ and the presentation of two least-squares methodologies for estimating the fixed parameter vector β in the model, and the description of connections between the two types of least-squares estimators and the reverse order laws for generalized inverses of AB. We then prepare various necessary matrix study tools, including a general theory on linear or nonlinear algebraic matrix identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for calculating the maximum and minimum ranks of B(s2,…,t2)A(s1,…,t1), as well as necessary and sufficient conditions for B(s2,…,t2)A(s1,…,t1) to be invariant with respect to the choice of the generalized inverses, etc. Subsequently, we present a unified approach to the 512 matrix set inclusion problems associated with the above reverse order laws for the eight commonly-used types of generalized inverses of A, B, and AB by means of the definitions of generalized inverses, the block matrix methodology (BMM), the matrix equation methodology (MEM), and the matrix rank methodology (MRM).

In this paper, we establish the solvability conditions and the formula of the general solution to a Sylvester-like quaternion matrix equation. As an application, we give some necessary and sufficient conditions for a system of quaternion matrix equations to be consistent, and present an expression of the general solution of the system when it is solvable. We present an algorithm and an example to illustrate the main results of this paper. The findings of this paper generalize the known results in the literature.

We in this paper define the outer-Perron-Frobenius splitting, which is an extension of the pseudo- Perron-Frobenius splitting defined in [A.N. Sushama, K. Premakumari, K.C. Sivakumar, Extensions of Perron-Frobenius splittings and relationships with nonnegative Moore-Penrose inverse, Linear and Multilinear Algebra 63 (2015) 1-11]. We present some criteria for the convergence of the outer-Perron-Frobenius splitting. The findings of this paper generalize some known results in the literatures.

Featured Application: The advantageous characteristics of flexibility and versatility of movement in spine biotensegrity models can be further studied for potential application in deployable structures and flexible arm in the robotic industry. This paper presents a study on form-finding of four-stage class one self-equilibrated spine biotensegrity models. Advantageous features such as slenderness and natural curvature of the human spine, as well as the stabilizing network that consists of the spinal column and muscles, were modeled and incorporated in the mathematical formulation of the spine biotensegrity models. Form-finding analysis, which involved determination of independent self-equilibrium stress modes using generalized inverse and their linear combination, was carried out. Form-finding strategy for searching the self-equilibrated models was studied through two approaches: application of various combinations of (1) twist angles and (2) nodal coordinates. A total of three configurations of the spine biotensegrity models with different sizes of triangular cell were successfully established for the first time in this study. All members in the spine biotensegrity models satisfied the assumption of linear elastic material behavior. With the established spine biotensegrity model, the advantageous characteristics of flexibility and versatility of movement can be further studied for potential application in deployable structures and flexible arm in the robotic industry. [ABSTRACT FROM AUTHOR]

In this paper, we derive a novel expression for the computation of the outer inverse A T , S (2) . Based on this expression, we present a new Gauss-Jordan elimination method for computing A R (G) , N (G) (2) . The analysis of computational complexity indicates that our algorithm is more efficient than the existing Gauss-Jordan elimination algorithms for A R (G) , N (G) (2) in the literature for a large class of problems. Especially for the case when G is a Hermitian matrix, our algorithm has the lowest computational complexity among the existing algorithms. Finally, numerical experiments show that our method for the outer inverse A R (G) , N (G) (2) generally is more efficient than that of the other existing methods in the cases of matrices A with m < n or square matrices G with high rank or Hermitian matrices G in practice. [ABSTRACT FROM AUTHOR]

In this paper, we define the direct sum relation on a module and prove that this is a partial order when the module is regular. We also focus on a regular support and a strong regular support of an element in a module, which are extensions of a generalized inverse and a reflexive generalized inverse of an element in a ring, respectively. Using regular supports and strong regular supports, we present various characterizations of the direct sum order as an application. We obtain some decompositions of a module via the direct sum order. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new representations and characterizations of the outer inverse of tensors through QR decomposition. Derived representations are usable in generating corresponding representations of main tensor generalized inverses. Some results on reshape operation of a tensor are added to the existing theory. An effective algorithm for computing outer inverses of tensors is proposed and applied. The power of the proposed method is demonstrated by its application in 3D color image deblurring. [ABSTRACT FROM AUTHOR]

Generalized inverses of a matrix product can be written as certain matrix expressions that are composed by the given matrices and their generalized inverses, and a challenging task in this respect is to establish various reasonable reverse order laws for generalized inverses of matrix products. In this paper, we present two groups of known and new mixed reverse order laws for the Moore–Penrose inverses of products of two and three matrices through various conventional matrix operations. We also establish four groups of matrix set inclusions that are composed by { 1 } - and { 1 , 2 } -generalized inverses of A, B, C, and their products AB and ABC. [ABSTRACT FROM AUTHOR]

Let T be a bounded linear operator from a Banach space to a Banach space with closed range and let T ¯ = T + δ T. Nashed's condition is that (I + δ T T +) − 1 T ¯ maps the null space of T into the range of T. The stable perturbation means that the intersection of the range of T ¯ and the null space of the generalized inverse of T is {0}. We show that the stable perturbation is the same as Nashed's condition in the sense of duality. [ABSTRACT FROM AUTHOR]

This note presents ad-nonprojection method for approximation to infinite-dimensional Moore-Penrose inverse, which includes generalized projection method as a special case. The main result (Theorem 4) gives characteristics of perfect convergence, weak perfect convergence, and uniform perfect convergence of ad-nonprojection method. This result is a generalization of several fundamental theorems known. [ABSTRACT FROM AUTHOR]

Let T be a bounded linear operator from a Banach space to a Banach space with closed range and let T ¯ = T + δ T. Nashed's condition is that (I + δ T T +) − 1 T ¯ maps the null space of T into the range of T. The stable perturbation means that the intersection of the range of T ¯ and the null space of the generalized inverse of T is {0}. We show that the stable perturbation is the same as Nashed's condition in the sense of duality. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the equality problem of generalized Bajraktarević means, i.e., to solve the functional equation * f (- 1) ( p 1 (x 1) f (x 1) + ⋯ + p n (x n) f (x n) p 1 (x 1) + ⋯ + p n (x n) ) = g (- 1) ( q 1 (x 1) g (x 1) + ⋯ + q n (x n) g (x n) q 1 (x 1) + ⋯ + q n (x n) ) , which holds for all (x 1 , ⋯ , x n) ∈ I n , where n ≥ 2 , I is a nonempty open real interval, the unknown functions f , g : I → R are strictly monotone, f (- 1) and g (- 1) denote their generalized left inverses, respectively, and p = (p 1 , ⋯ , p n) : I → R + n and q = (q 1 , ⋯ , q n) : I → R + n are also unknown functions. This equality problem in the symmetric two-variable (i.e., when n = 2 ) case was already investigated and solved under sixth-order regularity assumptions by Losonczi (Aequationes Math 58(3):223–241, 1999). In the nonsymmetric two-variable case, assuming the three times differentiability of f, g and the existence of i ∈ { 1 , 2 } such that either p i is twice continuously differentiable and p 3 - i is continuous on I, or p i is twice differentiable and p 3 - i is once differentiable on I, we prove that (*) holds if and only if there exist four constants a , b , c , d ∈ R with a d ≠ b c such that c f + d > 0 , g = a f + b c f + d , and q ℓ = (c f + d) p ℓ (ℓ ∈ { 1 , ⋯ , n }). In the case n ≥ 3 , we obtain the same conclusion with weaker regularity assumptions. Namely, we suppose that f and g are three times differentiable, p is continuous and there exist i , j , k ∈ { 1 , ⋯ , n } with i ≠ j ≠ k ≠ i such that p i , p j , p k are differentiable. [ABSTRACT FROM AUTHOR]

Some basic properties of the range and null space of multidimensional arrays (tensors) with respect to Einstein tensor product are investigated. Also, an useful and effective definition of the tensor rank, termed as reshaping rank, is introduced. Further, computation of tensor outer with prescribed range and/or kernel of higher order tensors is considered. Starting from an algebraic approach, some new relationships between the problems of solving tensor equations and computation of tensor outer inverses with prescribed range and/or null pace are established. Conditions for the existence, representation and computation of the Moore-Penrose inverse, the weighted Moore-Penrose inverse, the Drazin inverse and the usual inverse of tensors are derived as corollaries. In this way, we derive extensions of known representations of various classes of matrix generalized inverses to multidimensional arrays. Effective algorithms for computing inner inverses of tensors as well as initiated algorithms for computing outer inverses of tensors are presented. Implementation of proposed algorithms in Matlab are developed and illustrative numerical examples are given. In addition, results related with the (b , c) -inverses on semigroups are examined in details in a specific semigroup of tensors with a binary associative operation defined as the Einstein tensor product. [ABSTRACT FROM AUTHOR]

The notion of the core inverse of tensors with the Einstein product was introduced, very recently (Sahoo et al. Comp Appl Math 39:9, 2020). In this paper we establish a few sufficient and necessary conditions for reverse-order law of this inverse. Further, we discuss mixed-type reverse-order law for core inverse. In addition to these, we discuss core inverse solutions of multilinear systems via the Einstein product. The prowess of the inverse is demonstrated to solve the Poisson problem in the multilinear system framework. [ABSTRACT FROM AUTHOR]

In [Linear Algebra Appl. 436 (2012), 1909–1923], the author introduced, for any semigroup S and any a , b , c ∈ S , the (b, c)-invertibility of a as meaning the existence of some (b, c)-inverse y ∈ S such that y ∈ b S y ∩ y S c , yab = b and cay = c (and also introduced two other weaker version for rings), which generalizes both the Moore-Penrose inverse A † and the author's pseudo-inverse A ′ for square matrices A. The present article further generalizes (b, c)-inverses in two directions, first so as to apply to morphisms in arbitrary categories (rather than only to elements of semigroups) and secondly so as to yield weighted versions of (b, c)-inverses by simultaneously generalizing J. S. Chipman's weighted version of A † and R. E. Cline and T. N. E. Greville's weighted version of A ′. To enable some of these developments, H. Mitsch's well-known partial order in semigroups is extended to obtain a version applicable to morphisms in arbitrary categories. [ABSTRACT FROM AUTHOR]

This paper proposes a generalized fractional controller for integer order systems with time delay. The fractional controller structure is so adopted to have a combined effect of fractional filter and Smith predictor. Interestingly, the resulting novel controller can be decomposed into fractional filter cascaded with an integer order PID controller. The method is applied to two practical examples i.e. liquid level system and Shell control fractionator system. The closed- loop responses resulting from the proposed method are compared with that of the available methods in the literature. For quantitative evaluations of the proposed method, Integral Absolute Error (IAE) and Integral Square Control Input (ISCI) performance criteria are employed. The proposed method effectively enhances the closed-loop response by improving the IAE values, reducing the control effort inputs to achieve the desired output. The disturbance rejection and robustness tests are also carried out. The robustness test reveals a significant improvement in the maximum absolute sensitivity measure. That is displayed in numerical simulations of the paper. [ABSTRACT FROM AUTHOR]

Specific definitions of the core and core-EP inverses of complex tensors are introduced. Some characterizations, representations and properties of the core and core-EP inverses are investigated. The results are verified using specific algebraic approach, based on proposed definitions and previously verified properties. The approach used here is new even in the matrix case. [ABSTRACT FROM AUTHOR]

This paper is concerned with a linear quadratic (LQ) optimal control for discrete-time uncertain systems, with indefinite state and control weighting matrices in the cost function. Firstly, a recurrence equation of general optimal control problem for discrete-time uncertain systems is obtained by applying Bellman's principle of optimality. Then, the optimal state feedback control is obtained based on the recurrence equation. Moreover, a sufficient condition of well-posedness for the LQ problem is proposed and a general expression for the optimal control set is given. Furthermore, a numerical example is presented by using the obtained results. Finally, as an application of the indefinite LQ optimal control, an optimal production inventory problem of uncertain environment is solved. [ABSTRACT FROM AUTHOR]

Inverse problems are characterized by their inherent non-uniqueness and sensitivity with respect to data perturbations. Their stable solution requires the application of regularization methods including variational and iterative regularization methods. Superiorization is a heuristic approach that can steer basic iterative algorithms to have small value of a certain regularization functional while keeping the algorithms simplicity and computational efforts, and is able to account for additional prior information. In this note, we combine the superiorization methodology with iterative regularization and show that the superiorized version of the scheme yields again a regularization method, however accounting for different prior information. [ABSTRACT FROM AUTHOR]

In the theoretical studies and numerical computations of split quaternionic quantum mechanics, the singular value decomposition theorem of split quaternion matrices plays an essential role. In general the problem of singular value decomposition over split quaternion algebra has hitherto remained tangential for split quaternion matrices. In this paper, by means of a real representation matrix of a split quaternion matrix, the singular value decomposition of split quaternion matrices is studied, the singular value decomposition theorem and the corresponding algorithm for split quaternion matrices are given. Finally, three applications for solving the split quaternion least squares problem and the wave separation problem are given based on the singular value decomposition over split quaternion algebra. [ABSTRACT FROM AUTHOR]