Jacobson\u27s lemma for the generalized n-strong Drazin inverses in rings and in operator algebras

In this paper, we extend Jacobson\u27s lemma for Drazin inverses to the generalized (n)-strong Drazin inverses in a ring, and prove that (1-ac) is generalized (n)-strong Drazin invertible if and only if (1-ba) is generalized (n)-strong Drazin invertible, provided that (a(ba)^{2}=abaca=acaba=(ac)^{2}a). In addition, Jacobson\u27s lemma for the left and right Fredholm operators, and furthermore, for consistent in invertibility spectral property and consistent in Fredholm and index spectral property are investigated

Generalizations and Some Applications of Kronecker and Hadamard Products of Matrices

In this thesis, generalizations of Kronecker, Hadamard and usual products (sums) that depend on the partitioned of matrices are studied and defined. Namely: Tracy- Singh, Khatri-Rao, box, strong Kronecker, block Kronecker, block Hadamard, restricted Khatri-Rao products (sums) which are extended the meaning of Kronecker, Hadamard and usual products (sums). The matrix convolution products, namely: matrix convolution, Kronecker convolution and Hadamard convolution products of matrices with entries in set of functions are also considered. The connections among them are derived and most useful properties are studied in order to find new applications of Tracy-Singh and Khatri-Rao products (sums). These applications are: a family of generalized inverses, a family of coupled singular matrix problems, a family of matrix inequalities and a family of geometric means. In the theory of generalized inverses of matrices and their applications, the five generalized inverses, namely Moore-Penrose, weighted Moore-Penrose, Drazin, weighted Drazin and group inverses and their expressions and properties are studied. Moreover, some new numerous matrix expressions involving these generalized inverses and weighted matrix norms of the Tracy-Singh products matrices are also derived. In addition, we establish some necessary and sufficient conditions for the reverse order law of Drazin and weighted Drazin inverses. These results play a central role in our applications and many other applications. In the field of system identification and matrix products work, we propose several algorithms for computing the solutions of the coupled matrix differential equations, coupled matrix convolution differential, coupled matrix equations, restricted coupled singular matrix equations, coupled matrix least-squares problems and weighted Least -squares problems based on idea of Kronecker (Hadamard) and Tracy-Singh(Khatri-Rao) products (sums) of matrices. The way exists which transform the coupled matrix problems and coupled matrix differential equations into forms for which solutions may be readily computed. The common vector exact solutions of these coupled are presented and, subsequently, construct a computationally - efficient solution of coupled matrix linear least-squares problems and nonhomogeneous coupled matrix differential equations. We give new applications for the representations of weighted Drazin, Drazin and Moore-Penrose inverses of Kronecker products to the solutions of restricted singular matrix and coupled matrix equations. The analysis indicates that the Kronecker (Hadamard) structure method can achieve good efficient while the Hadamard structure method achieve more efficient when the unknown matrices are diagonal. Several special cases of these systems are also considered and solved, and then we prove the existence and uniqueness of the solution of each case, which includes the well-known coupled Sylvester matrix equations. We show also that the solutions of non-homogeneous matrix differential equations can be written in convolution forms. The analysis indicates also that the algorithms can be easily to find the common exact solutions to the coupled matrix and matrix differential equations for partitioned matrices by using the connections between Tracy-Singh, Block Kronecker and Khatri -Rao products and partitioned vector row (column) and our definition which is the so-called partitioned diagonal extraction operators. Unlike Matrix algebra, which is based on matrices, analysis must deal with estimates. In other words, Inequalities lie at the core of analysis. For this reason, it’s of great importance to give bounds and inequalities involving matrices. In this situation, the results are organized in the following five ways: First, we find some extensions and generalizations of the inequalities involving Khatri-Rao products of positive (semi) definite matrices. We turn to results relating Khatri-Rao and Tracy- Singh powers and usual powers, extending and generalizing work of previous authors. Second, we derive some new attractive inequalities involving Khatri-Rao products of positive (semi) definite matrices. We remark that some known inequalities and many other new interesting inequalities can easily be found by using our approaches. Third, we study some sufficient and necessary conditions under which inequalities below become equalities. Fourth, some counter examples are considered to show that some inequalities do not hold in general case. Fifth, we find Hölder-type inequalities for Tracy-Singh and Khatri-Rao products of positive (semi) definite matrices. The results lead to inequalities involving Hadamard and Kronecker products, as a special case, which includes the well-known inequalities involving Hadamard product of matrices, for instance, Kantorovich-type inequalities and generalization of Styan's inequality. We utilize the commutativity of the Hadamard product (sum) for possible to develop and improve some interesting inequalities which do not follow simply from the work of researchers, for example, Visick's inequality. Finally, a family of geometric means for positive two definite matrices is studied; we discuss possible definitions of the geometric means of positive definite matrices. We study the geometric means of two positive definite matrices to arrive the definitions of the weighted operator means of positive definite matrices. By means of several examples, we show that there is no known definition which is completely satisfactory. We have succeeded to find many new desirable properties and connections for geometric means related to Tracy-Singh products in order to obtain new unusual estimates for the Khatri-Rao (Tracy-Singh) products of several positive definite matrices

Different invertibility modifications in operator spaces and c*-algebras and its applications

In this thesis different modifications of invertibility in various settings and their applications are investigated. In particular, the reverse order law is considered for classes of {1,3} and {1,4}-generalized inverses in C*-algebras and particulary in the vector space of linear bounded operators on separable Hilbert spaces. The Hartwig's triple reverse order law for Moore-Penrose inverse is discussed in C*-algebra and ring with involution settings. The reverse order laws on {1,3}, {1,4}, {1,3,4}, {1,2,3} and {1,2,4}-inverses in a ring setting are investigated. This results contain improvements of some known results in C*-algebra case because the assumptions of the regularity of some elements are omitted. The generalized invertibility is applied to solving certain types of equations in rings with unit and determining the general form of solutions. Strictly, the algebraic conditions for the existence of a solution and the expression for the general solution of the system of three linear equations in a ring with a unit are discussed. Another research concerns when the linear combinations of two operators belonging to the class of Fredholm operators. Some cases where the Fredholmness of linear combination is independent of the choice of the scalars are described in detail

THE OPTIMAL PROJECTION EQUATIONS FOR FINITE-DIMENSIONAL FIXED-ORDER DYNAMIC COMPENSATION OF INFINITE-DIMENSIONAL SYSTEMS

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/57874/1/OptimalProjInfDiml1986.pd

Functional Calculus

The aim of this book is to present a broad overview of the theory and applications related to functional calculus. The book is based on two main subject areas: matrix calculus and applications of Hilbert spaces. Determinantal representations of the core inverse and its generalizations, new series formulas for matrix exponential series, results on fixed point theory, and chaotic graph operations and their fundamental group are contained under the umbrella of matrix calculus. In addition, numerical analysis of boundary value problems of fractional differential equations are also considered here. In addition, reproducing kernel Hilbert spaces, spectral theory as an application of Hilbert spaces, and an analysis of PM10 fluctuations and optimal control are all contained in the applications of Hilbert spaces. The concept of this book covers topics that will be of interest not only for students but also for researchers and professors in this field of mathematics. The authors of each chapter convey a strong emphasis on theoretical foundations in this book

Index and characteristic analysis of partial differential equations

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2000.Includes bibliographical references (leaves 230-238) and index.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Technologies for dynamic simulation of chemical process flowsheets, as implemented in equation-based dynamic simulators,allow solution of fairly sophisticated process models, that include detailed descriptions of physical phenomena along with operating policies and discrete events. Simulation of flowsheet models with this level of detail forms the basis for a wide variety of activities, such as process optimization, startup and shutdown studies, process design, batch policy synthesis, safety interlock validation, and operator training. Technologies that make these activities possible for plant-scale models include sparse linear algebra routines, robust backward difference formula time integration methods, guaranteed state event location algorithms, generation of analytical Jacobian information via automatic differentiation, efficient algorithms for consistent initialization that may also be used to analyze the index of the model equations, automatic index reduction algorithms, and path-constrained dynamic optimization methods. An equation-based dynamic process simulator takes as input the model equations that describe process behavior, along with a description of the operating policy. The input language allows for model decomposition, inheritance, and reuse, which facilitates construction of plant-scale dynamic models. Technologies like the ones mentioned above allow the simulator to then analyze the model for inconsistencies and perform calculations based on dynamic simulation, with a minimum of intervention from the engineer. This reduces both the time and numerical expertise required to perform simulation-based activities. Results, in some cases made possible or economically feasible only by the modeling support provided by a simulator,(cont.) have been impressive. However, these capabilities apply to flowsheet models that consist only of differential-algebraic, or lumped, unit models. Sometimes behavior in a particular unit cannot be adequately described by a lumped formulation, when variation with other independent variables like distance along a PFTR, film coordinate, or polymer chain length are important. In this case, behavior is most naturally modeled with partial differential, or distributed, unit models. Partial differential equations in network flow simulations bring an additional set of mathematical and numerical issues. For a distributed model to bema thematically well-posed, proper initial and boundary conditions must be specified. Boundary condition requirements for nonlinear unit models may change during the course of a dynamic simulation, even in the absence of discrete events. Some distributed models, due to improper formulation or simple transcription errors, may be ill-posed because they do not have a mathematical property called continuous dependence on data. Finally, the model equations must be discretized in the proper manner. This thesis contributes two new analyses of distributed unit models. The first relies on the definition of a differentiation index for partial differential equations developed in this thesis. It is by design a very natural generalization of the differentiation index of differential-algebraic equations.by Wade Steven Martinson.Ph.D