Left Inverses Research Articles

Left w-Core Inverses in Rings with Involution

Left w-Core Inverses in Rings with Involution

A Deterministic Least Squares Approach for Simultaneous Input and State Estimation

This paper considers a deterministic estimation problem to find the input and state of a linear dynamical system which minimise a weighted integral squared error between the resulting output and the measured output. A completion of squares approach is used to find the unique optimum in terms of the solution of a Riccati differential equation. The optimal estimate is obtained from a two-stage procedure that is reminiscent of the Kalman filter. The first stage is an end-of-interval estimator for the finite horizon which may be solved in real time as the horizon length increases. The second stage computes the unique optimum over a fixed horizon by a backwards integration over the horizon. A related tracking problem is solved in an analogous manner. Making use of the solution to both the estimation and tracking problems a constrained estimation problem is solved which shows that the Riccati equation solution has a least squares interpretation that is analogous to the meaning of the covariance matrix in stochastic filtering. The paper shows that the estimation and tracking problems considered here include the Kalman filter and the linear quadratic regulator as special cases. The infinite horizon case is also considered for both the estimation and tracking problems. Stability and convergence conditions are provided and the optimal solutions are shown to take the form of left inverses of the original system.

Algebras with a bilinear form, and idempotent endomorphisms

The category k-BFAlg of all k-algebras with a bilinear form, whose objects are all pairs (R,b) where R is a k-algebra and b:R×R→k is a bilinear mapping, is equivalent to the category of unital k-algebras A for which the canonical homomorphism (k,1)→(A,1A) of unital k-algebras is a splitting monomorphism in k-Mod. Call the left inverses of this splitting monomorphism “weak augmentations” of the algebra. There is a category isomorphism between the category of k-algebras with a weak augmentation and the category of unital k-algebras (A,bA) with a bilinear form bA compatible with the multiplication of A, i.e., such that bA(x,y)=bA(z,w) for all x,y,z,w∈A for which xy=zw.

Weakly concave operators

We study a class of left-invertible operators which we call weakly concave operators. It includes the class of concave operators and some subclasses of expansive strict $m$-isometries with $m > 2$. We prove a Wold-type decomposition for weakly concave operators. We also obtain a Berger–Shaw-type theorem for analytic finitely cyclic weakly concave operators. The proofs of these results rely heavily on a spectral dichotomy for left-invertible operators. It provides a fairly close relationship, written in terms of the reciprocal automorphism of the Riemann sphere, between the spectra of a left-invertible operator and any of its left inverses. We further place the class of weakly concave operators, as the term $\mathcal {A}_1$, in the chain $\mathcal {A}_0 \subseteq \mathcal {A}_1 \subseteq \ldots \subseteq \mathcal {A}_{\infty }$ of collections of left-invertible operators. We show that most of the aforementioned results can be proved for members of these classes. Subtleties arise depending on whether the index $k$ of the class $\mathcal {A}_k$ is finite or not. In particular, a Berger–Shaw-type theorem fails to be true for members of $\mathcal {A}_{\infty }$. This discrepancy is better revealed in the context of $C^*$- and $W^*$-algebras.

Two unified families of bivariate Mittag-Leffler functions

The various bivariate Mittag-Leffler functions existing in the literature are gathered here into two broad families. Several different functions have been proposed in recent years as bivariate versions of the classical Mittag-Leffler function; we seek to unify this field of research by putting a clear structure on it. We use our general bivariate Mittag-Leffler functions to define fractional integral operators (which have a semigroup property) and corresponding fractional derivative operators (which act as left inverses and analytic continuations). We also demonstrate how these functions and operators arise naturally from some fractional partial integro-differential equations of Riemann–Liouville type.

Left and right-Drazin inverses in rings and operator algebras

The paper introduces the left and right versions of the large class of Drazin inverses in terms of the left and right annihilators in a ring, which are called left-Drazin and right-Drazin inverses. We characterize some basic properties of these one-sided Drazin inverses, and discuss Jacobson’s lemma for them. In addition, the relation between the Drazin inverses and these two one-sided inverses is given by means of the spectrum and the operator decomposition. As an application, the left-Drazin and right-Drazin invertibilities in the Calkin algebra are investigated.

Left core inverses in rings with involution

Left core inverses in rings with involution

Classification of quasigroups according to directions of translations II

In each quasigroup $Q$ there are defined six types of translations: the left, right and middle translations and their inverses. Two translations may coincide as permutations of $Q$, and yet be different when considered upon the web of the quasigroup. We shall call each of the translation types a direction and will associate it with one of the elements $\iota, l, r, s, ls $ and $rs$, i.e., the elements of a symmetric group $S_3$. Properties of the directions are considered in part 1 of "Classification of quasigroups according to directions of translations I" by F. M. Sokhatsky and A. V. Lutsenko. Let ${^{\sigma}\mathcal{M}}$ denote the set of all translations of a direction $\sigma\in S_{3}$. The conditions ${^{\sigma}\mathcal{M}}={^{\kappa}\mathcal{M}}$, where $\sigma,\kappa\in S_{3}$ and $\sigma\ne\kappa$, define nine quasigroup varieties. Four of them are well known: $LIP$, $RIP$, $MIP$ and $CIP$. The remaining five quasigroup varieties are relatively new because they are left and right inverses of $ CIP$ variety and generalization of commutative, left and right symmetric quasigroups.

Abelian and symmetric generalized digroups

In this paper we extend some group-like concepts to generalized digroups. We exhibit examples of non-abelian cyclic generalized digroups. We show that abelian generalized digroups are tightly related to left commutative semigroups with left identities and right inverses, and characterize these semigroups using generalized digroups properties. We also present symmetric generalized digroups and the transformation generalized digroup.

Left and right G-outer inverses

The main contribution of this paper is to present new generalized inverses as weaker versions of a G-outer inverse. In particular, we define and characterize left and right G-outer inverses of rectangular matrices. Solvability of matrix equation systems as AXA = AEA and BAEAX = B; or AXA = AEA and XAEAD = D, where , , and , is studied by means of left and right G-outer inverses. The general solution forms of these systems give descriptions of the sets of all left and right G-outer inverses. Using left and right G-outer inverses, we introduce new partial orders and establish their relations with minus partial order and space pre-order. We apply these results to present and investigate left and right G-Drazin inverses of square matrices and corresponding partial orders.

Weighted quaternion core-EP, DMP, MPD, and CMP inverses and their determinantal representations

In this paper, we extend notions of the weighted core-EP right and left inverses, the weighted DMP and MPD inverses, and the CMP inverse to matrices over the quaternion skew field that have some features in comparison to these inverses over the complex field. We give the direct methods of their computing, namely, their determinantal representations by using noncommutative column and row determinants previously introduced by the author. A numerical example to illustrate the main result is given.

Determinantal Representations of the Weighted Core-EP, DMP, MPD, and CMP Inverses

In this paper, new notions of the weighted core-EP left inverse and the weighted MPD inverse which are dual to the weighted core-EP (right) inverse and the weighted DMP inverse, respectively, are introduced and represented. The direct methods of computing the weighted right and left core-EP, DMP, MPD, and CMP inverses by obtaining their determinantal representations are given. A numerical example to illustrate the main result is given.

Algebraic Inverses on Lie Algebra Comultiplications

In this note, we investigate algebraic loop structures and inverses of elements of a set of all homomorphisms of Lie algebras with a binary operation derived from a Lie algebra comultiplication. As a symmetry phenomenon, we show that if l ( 1 ) c and r ( 1 ) c are the left and right inverses of the identity 1 : L → L on a free graded Lie algebra L , respectively, based on the Lie algebra comultiplication ψ c : L → L ⊔ L , then we have l ( 1 ) = l ( 1 ) c and r ( 1 ) = r ( 1 ) c , where c : L → L ⊔ L is a commutator.

Scalar-type kernels for block Toeplitz operators

It is shown that the kernel of a Toeplitz operator with 2×2 symbol G can be described exactly in terms of any given function in a very wide class, its image under multiplication by G, and their left inverses, if the latter exist. As a consequence, under many circumstances the kernel of a block Toeplitz operator may be described as the product of a space of scalar complex-valued functions by a fixed column vector of functions. Such kernels are said to be of scalar type, and in this paper they are studied and described explicitly in many concrete situations. Applications are given to the determination of kernels of truncated Toeplitz operators for several new classes of symbols.

Minimal-delay FIR delayed left inverses for systems with zero nonzero zeros

This paper considers finite-time input reconstruction for discrete-time linear time-invariant systems in the case where the initial condition is unknown. There are three main results. First, a specific construction of finite-impulse-response (FIR) delayed left inverse with the minimal delay for systems with zero nonzero zeros is presented. Next, it is shown that, in the presence of an arbitrary unknown initial condition, finite-time input reconstruction is possible using a delayed left inverse H if and only if H is FIR. Finally, it is shown that a transfer function with full column normal rank has an FIR delayed left inverse with the minimal delay if and only if the system has zero nonzero zeros.

Determinantal Representations of the Core Inverse and Its Generalizations with Applications

In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equations that is an analog of Cramer’s rule. A numerical example to illustrate the main result is given.

On deterministic weighted automata

Two families of input-deterministic weighted automata over semirings are considered: purely sequential automata, in which terminal weights of states are either zero or unity, and sequential automata, in which states can have arbitrary terminal weights. The class of semirings over which all weighted automata admit purely sequential equivalents is fully characterised. A similar characterisation is proved for sequential automata under an assumption that all elements of the underlying semiring have finitely many multiplicative left inverses, which is in particular true for all commutative semirings and all division semirings.

The eigenvalues and eigenvectors of nonsingular tensors, similar tensors and tensor products

In this paper, we extend some well-known results on the relationships of the eigenvalues and eigenvectors from matrices to the tensors case, including nonsingular tensors and their left (right) inverses, similar tensors, tensor products and , where B is a matrix.

Retractions in Intersection Types

This paper deals with retraction - intended as isomorphic embedding - in intersection types building left and right inverses as terms of a lambda calculus with a bottom constant. The main result is a necessary and sufficient condition two strict intersection types must satisfy in order to assure the existence of two terms showing the first type to be a retract of the second one. Moreover, the characterisation of retraction in the standard intersection types is discussed.

Absolute retractness of automata on directed complete posets

The notion of retractness, which is about having left inverses (reflection) for monomorphisms, is crucial in most branches of mathematics. One very important notion related to it is injectivity, which is about extending morphisms to larger domains and plays a fundamental role in many areas of mathematics as well as in computer science, under the name of complete or partial objects. Absolute retractness is tightly related to injectivity and is in fact equivalent to it in many categories. In this paper, combining the two important notions of actions of semigroups and directed complete posets, which are both crucial abstraction and useful in mathematics as well as in computer science, we consider the category Dcpo-[Formula: see text] of actions of a directed complete semigroup on directed complete posets, and study absolute retractness with respect to both monomorphisms and embeddings in this category. Among other things, we show that absolute retract ([Formula: see text]-)dcpo’s are complete but the converse is not necessarily true. Investigating the converse, we find that if we add the property of being a countable chain to completeness, over some kinds of dcpo-monoids such as dcpo-groups and commutative monoids, we get absolute retractness. Furthermore, we show that there are absolute retract [Formula: see text]-dcpo’s, which are not chains.