On the weighted reverse order laws for the Moore-Penrose inverse and K-inverses

The main objective of this article is to study several generalizations of the reverse order law for the Moore-Penrose inverse in ring with involution.Comment: 13 pages, original research articl

Further results on the Drazin inverse of even-order tensors

The notion of the Drazin inverse of an even-order tensor with the Einstein product was introduced, very recently [J. Ji and Y. Wei. Comput. Math. Appl., 75(9), (2018), pp. 3402-3413]. In this article, we further elaborate this theory by producing a few characterizations of the Drazin inverse and the W-weighted Drazin inverse of tensors. In addition to these, we compute the Drazin inverse of tensors using different types of generalized inverses and full rank decomposition of tensors. We also address the solution to the multilinear systems using the Drazin inverse and iterative (higher order Gauss-Seidel) method of tensors. Besides this, the convergence analysis of the iterative technique is also investigated within the framework of the Einstein product.Comment: 3

Reverse order laws for generalized inverses of products of two or three matrices with applications

One of the fundamental research problems in the theory of generalized inverses of matrices is to establish reverse order laws for generalized inverses of matrix products. Under the assumption that $A$, $B$, and $C$ are three nonsingular matrices of the same size, the products $AB$ and $ABC$ are nonsingular as well, and the inverses of $AB$ and $ABC$ admit the reverse order laws $(AB)^{-1} = B^{-1} A^{-1}$ and $(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$, respectively. If some or all of $A$, $B$, and $C$ are singular, two extensions of the above reverse order laws to generalized inverses can be written as $(AB)^{(i,\ldots,j)} = B^{(i_2,\ldots,j_2)} A^{(i_1,\ldots,j_1)}$ and $(ABC)^{(i,\ldots,j)} = C^{(i_3,\ldots,j_3)} B^{(i_2,\ldots,j_2)}A^{(i_1,\ldots,j_1)}$, or other mixed reverse order laws. These equalities do not necessarily hold for different choices of generalized inverses of the matrices. Thus it is a tremendous work to classify and derive necessary and sufficient conditions for the reverse order law to hold because there are all 15 types of $\{i,\ldots, j\}$-generalized inverse for a given matrix according to combinatoric choices of the four Penrose equations. In this paper, we first establish several decades of mixed reverse order laws for $\{1\}$- and $\{1,2\}$-generalized inverses of $AB$ and $ABC$. We then give a classified investigation to a special family of reverse order laws $(ABC)^{(i,\ldots,j)} = C^{-1}B^{(k,\ldots,l)}A^{-1}$ for the eight commonly-used types of generalized inverses using definitions, formulas for ranges and ranks of matrices, as well as conventional operations of matrices. Furthermore, the special cases $(ABA^{-1})^{(i,\ldots,j)} = AB^{(k,\ldots,l)}A^{-1}$ are addressed and some applications are presented.Comment: 2

Rank Equalities Related to Generalized Inverses of Matrices and Their Applications

This paper is divided into two parts. In the first part, we develop a general method for expressing ranks of matrix expressions that involve Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose inverses of matrices. Through this method we establish a variety of valuable rank equalities related to generalized inverses of matrices mentioned above. Using them, we characterize many matrix equalities in the theory of generalized inverses of matrices and their applications. In the second part, we consider maximal and minimal possible ranks of matrix expressions that involve variant matrices, the fundamental work is concerning extreme ranks of the two linear matrix expressions $A - BXC$ and $A - B_1X_1C_1 - B_2X_2C_2$. As applications, we present a wide range of their consequences and applications in matrix theory.Comment: 245 pages, LaTe

The (b,c)(b, c)-inverse in rings and in the Banach context

In this article the $(b, c)$-inverse will be studied. Several equivalent conditions for the existence of the $(b,c)$-inverse in rings will be given. In particular, the conditions ensuring the existence of the $(b,c)$-inverse, of the annihilator $(b,c)$-inverse and of the hybrid $(b,c)$-inverse will be proved to be equivalent, provided $b$ and $c$ are regular elements in a unitary ring $R$. In addition, the set of all $(b,c)$-invertible elements will be characterized and the reverse order law will be also studied. Moreover, the relationship between the $(b,c)$-inverse and the Bott-Duffin inverse will be considered. In the context of Banach algebras, integral, series and limit representations will be given. Finally the continuity of the $(b,c)$-inverse will be characterizedComment: 20 pages, original research articl

An online parameter identification method for time dependent partial differential equations

Online parameter identification is of importance, e.g., for model predictive control. Since the parameters have to be identified simultaneously to the process of the modeled system, dynamical update laws are used for state and parameter estimates. Most of the existing methods for infinite dimensional systems either impose strong assumptions on the model or cannot handle partial observations. Therefore we propose and analyze an online parameter identification method that is less restrictive concerning the underlying model and allows for partial observations and noisy data. The performance of our approach is illustrated by some numerical experiments

Krylov subspace methods and their generalizations for solving singular linear operator equations with applications to continuous time Markov chains

Viele Resultate über MR- und OR-Verfahren zur Lösung linearer Gleichungssysteme bleiben (in leicht modifizierter Form) gültig, wenn der betrachtete Operator nicht invertierbar ist. Neben dem für reguläre Probleme charakteristischen Abbruchverhalten, kann bei einem singulären Gleichungssystem auch ein so genannter singulärer Zusammenbruch auftreten. Für beide Fälle werden verschiedene Charakterisierungen angegeben. Die Unterrauminverse, eine spezielle verallgemeinerte Inverse, beschreibt die Näherungen eines MR-Unterraumkorrektur-Verfahrens. Für Krylov-Unterräume spielt die Drazin-Inverse eine Schlüsselrolle. Bei Krylov-Unterraum-Verfahren kann a-priori entschieden werden, ob ein regulärer oder ein singulärer Abbruch auftritt. Wir können zeigen, dass ein Krylov-Verfahren genau dann für beliebige Startwerte eine Lösung des linearen Gleichungssystems liefert, wenn der Index der Matrix nicht größer als eins und das Gleichungssystem konsistent ist. Die Berechnung stationärer Zustandsverteilungen zeitstetiger Markov-Ketten mit endlichem Zustandsraum stellt eine praktische Aufgabe dar, welche die Lösung eines singulären linearen Gleichungssystems erfordert. Die Eigenschaften der Übergangs-Halbgruppe folgen aus einfachen Annahmen auf rein analytischem und matrixalgebrischen Wege. Insbesondere ist die erzeugende Matrix eine singuläre M-Matrix mit Index 1. Ist die Markov-Kette irreduzibel, so ist die stationäre Zustandsverteilung eindeutig bestimmt

The entropy production of stationary diffusions

The entropy production rate is a central quantity in non-equilibrium statistical physics, scoring how far a stochastic process is from being time-reversible. In this paper, we compute the entropy production of diffusion processes at non-equilibrium steady-state under the condition that the time-reversal of the diffusion remains a diffusion. We start by characterising the entropy production of both discrete and continuous-time Markov processes. We investigate the time-reversal of time-homogeneous stationary diffusions and recall the most general conditions for the reversibility of the diffusion property, which includes hypoelliptic and degenerate diffusions, and locally Lipschitz vector fields. We decompose the drift into its time-reversible and irreversible parts, or equivalently, the generator into symmetric and antisymmetric operators. We show the equivalence with a decomposition of the backward Kolmogorov equation considered in hypocoercivity theory, and a decomposition of the Fokker-Planck equation in GENERIC form. The main result shows that when the time-irreversible part of the drift is in the range of the volatility matrix (almost everywhere) the forward and time-reversed path space measures of the process are mutually equivalent, and evaluates the entropy production. When this does not hold, the measures are mutually singular and the entropy production is infinite. We verify these results using exact numerical simulations of linear diffusions. We illustrate the discrepancy between the entropy production of non-linear diffusions and their numerical simulations in several examples and illustrate how the entropy production can be used for accurate numerical simulation. Finally, we discuss the relationship between time-irreversibility and sampling efficiency, and how we can modify the definition of entropy production to score how far a process is from being generalised reversible.Comment: 27 pages of main text, 7 figures, 43 pages including appendix and reference

Mathematical Aspects of General Relativity (hybrid meeting)

General relativity is an area that naturally combines differential geometry, partial differential equations, global analysis and dynamical systems with astrophysics, cosmology, high energy physics, and numerical analysis. It is rapidly expanding and has witnessed remarkable developments in recent years