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In the paper, we investigate the commutativity of a two-torsion free prime ring R provided with generalized derivations, and some well-known results that characterize the commutativity of prime rings through generalized derivations have been generalized. Moreover, we provide some examples to testify that the assumed restriction in our theorems cannot be omitted. [ABSTRACT FROM AUTHOR]

In this paper, commutativity of the compressions of kth order slant Toeplitz operators is discussed. We show that the commutativity and essential commutativity of two compressions of kth order slant Toeplitz operators on H 2 are same. We also establish some results on the compactness of the compressions of kth order slant Toeplitz operators. In the last section, we discuss various similarities and differences between compactness and non-compactness of these operators by simply looking at their graphs. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to initiate a new approach consisting to characterize the commutativity of a factor ring R/P with respect to a prime ideal P of a ring R admits some central valued identities with derivations of R. Some well-known results concerning the commutativity of prime rings have been improved. Moreover we provide examples to show that various restrictions in the hypotheses of our theorems are not superfluous. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a 3-prime near ring N involving multiplicative derivation d : N ⟶ N satisfying certain identities is a commutative ring. Also, an example is given to show that the necessity of the 3-primeness hypothesis imposed on the various theorems cannot be marginalized. [ABSTRACT FROM AUTHOR]

This paper investigates the commutativity of Banach algebras in which the group of invertible elements is subject to certain algebraic and metric conditions. Application to a particular class of Banach algebras is examined. [ABSTRACT FROM AUTHOR]

In this paper, we mainly show that if a product AB (or BA) of a closed symmetric operator A and a bounded positive operator B is normal, then it is self-adjoint. Equivalently, this means that B commutes with A. Certain generalizations and consequences are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we show that some additive mappings of a Banach algebra are derivations. Moreover, we exhibit some commutativity criteria for a prime Banach algebra admitting a derivation satisfying certain algebraic identities. [ABSTRACT FROM AUTHOR]

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Let A be a Banach algebra and n > 1 , a fixed integer. The main objective of this paper is to talk about the commutativity of Banach algebras via its projections. Precisely, we prove that if A is a prime Banach algebra admitting a continuous projection P with image in Z (A) such that P (a n) = a n for all a ∈ G , the nonvoid open subset of A , then A is commutative and P is the identity mapping on A . Apart from proving some other results, as an application we prove that, a normed algebra is commutative iff the interior of its center is non-empty. Furthermore, we provide some examples to show that the assumed restrictions cannot be relaxed. Finally, we conclude our paper with a direction for further research. [ABSTRACT FROM AUTHOR]

In this paper, we will extend some results on the commutativity of Jordan ideals proved in [8] and [5]. However, instead of left generalized derivations, we will consider generalized left derivations, which are sufficient to obtain good results with respect to the structure of near-rings. We will also show that the conditions imposed in the paper cannot be removed. [ABSTRACT FROM AUTHOR]

Let U be an invariant subset of finite Γ-near-ring M. There are many papers that consider the graph respect to the near-ring and the interplay between algebraic structures and graphs are studied. Indeed, it is worthwhile to relate algebraic properties of near-ring to the combinatorics properties of assigned graphs. In this paper the graph with respect to an invariant subset U of Γ-near-ring M, denoted by ΓαU(M) is introduced and the basic properties of it is investigated. Also the relation between the commutativity of M and properties of this graph is presented. [ABSTRACT FROM AUTHOR]

Our objective in this paper is to study the structure of 3-prime nearrings satisfying some algebraic properties. [ABSTRACT FROM AUTHOR]

In this study, necessary and sufficient conditions for the decomposition of any second-order discrete-time linear time-varying system as a commutative pair of two first-order systems are presented. Commutativity conditions with zero initial conditions are first expressed by Theorem 1. Then, the conditions under nonzero initial conditions are studied and presented by Theorem 2. The results including decomposition formulas are well verified by examples worked by using MATLAB Simulink tool. The importance of the paper subject is emphasized in view of some engineering applications for getting better performance characteristics such as sensitivity, robustness and stability. In fact, the results of this paper can be used when the synthesis of a second-order system is made by cascaded and commutative pairs of simple first-order subsystems; and by using the explicit decomposition formulas derived in the paper this can be achieved very easily. Since different decompositions lead to different performance characteristics, one should choose the appropriate ones among them. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to demonstrate that a quotient ring S/G is commutative by studying the behavior of symmetric generalized bi-derivations of S × S that fulfill some algebraic identities concerning a prime ideal. The result of Vukman is extended by proposing identities dealing with different symmetric generalized bi-additive mappings: for all elements l1 of a ring S, the statement λ(l1, l1)l1 + l1λ(l1, l1) - λ(l1, l1) belongs to a prime ideal. [ABSTRACT FROM AUTHOR]

In this paper, we introduce commutative, medial, normal, irreducible and prime filters in eBE-algebras, and investigate some of them properties. We define and study the concept of commutative eBE-algebra. In particular, we show that every commutative eBE-algebra is a commutative BE-algebra and prove that each filter of such algebra is commutative. Moreover, we give a construction of quotient eBE-algebras via normal filters. [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]

In this paper, we investigate permutation identities satisfied by semigroup left ideals and weak semigroup left ideals in prime nearrings. We obtain results on commutativity of multiplication and addition using these identities. We provide examples to show the necessity of certain conditions in some of the results obtained. Finally, we give a characterization of Galois field in terms of permutation identities. [ABSTRACT FROM AUTHOR]

In the present paper, we prove some commutativity theorems for a prime ring with involution in which generalized derivations satisfy certain differential identities. Some well known results on commutativity of prime rings have been obtained. Also, we provide an example to show that the assumed restriction imposed on the involution is not superfluous. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study certain central-valued identities on a factor ring with respect to a prime ideal of a ring ℛ , involving some specific mappings of ℛ. In addition, examples are given to demonstrate that the constraints imposed in the assumptions of our theorems are not superfluous. [ABSTRACT FROM AUTHOR]

There is a large body of evidence showing that the existence of a suitably-constrained derivation on a 3-prime near-ring forces the near-ring to be a commutative ring. In the present paper, we investigate the notion of left generalized derivation satisfying certain algebraic identities in 3-prime near-ring N which forces N to be a noncommutative ring. Moreover, an example proving the necessity of the primeness of N is given. [ABSTRACT FROM AUTHOR]

This article aims to create commutative hyperstructures, starting with a non-commutative group. Therefore, we consider the starting group to be the dihedral group Dn, where n is a natural number, n > 1, and we determine the HX groups associated with the dihedral group. For a fixed number n, we note Gn={Gp2p1 HX - groups, for any p1, p2 ∈ N* such that n = p1p2} as the set of all HX groups. This paper analyses this new structure's properties for particular cases when the dihedral group D4 is the support group. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study derivations and generalized derivations in prime rings satisfying certain differential identities. Some wellknown results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed. [ABSTRACT FROM AUTHOR]

Summary: The concept of full commutativity of displacements in the expression for strain energy density for the geometrically nonlinear problem has been introduced for the first time and fully established in this paper. Its consequences for the FEM formulation have been demonstrated. As a result, the strain energy, equilibrium equation, and incremental equilibrium equation for the geometrically nonlinear problem can all be presented in a unified manner involving various stiffness matrices that are all symmetric, unique, and explicitly expressed. As an important application, the framework has been employed in the FEM implementation of Koiter's initial postbuckling theory, which has been handicapped by its mesh sensitivity in evaluating one of the initial postbuckling coefficients. This has largely prevented it from being incorporated in mainstream commercial FEM codes. Based on the outcomes of this paper, the mesh sensitivity problem has been completely resolved without the need to use any specially formulated element. As a result, Koiter's theory can be practically and straightforwardly implemented in any FEM code. The results have been verified against those found in the literature. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to scrutinize the deportment of generalized derivations of R satisfying some functional * {*} -identities involving the center of the factor ring R / P {R/P} where P is a prime ideal of the ring R. Moreover, we suggest to give generalization of some well known results. [ABSTRACT FROM AUTHOR]

The main purpose of the present paper is to study the commutativity of a quotient ring R/P by considering derivations of R satisfying algebraic identities involving the prime ideal P. This approach allows us to generalize some well known results characterizing commutativity of rings. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is to connect the commutativity of a quotient ring R/P with endomorphisms of R satisfying certain algebraic identities involving prime ideals. The obtained results cover many known theorems (cf. Bell and Daif in Canad Math Bull 37(4):443–447, 1994; Deng and Ashraf in Results Math 30(3–4):259–263, 1996; Oukhtite et al. in Int Electron J Algebra 28:127–140, 2020). [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the commutativity of a prime ring R with involution of the second kind provided with two endomorphisms satisfying certain algebraic identities. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous. [ABSTRACT FROM AUTHOR]

In this paper, the notion of multiplicative generalized (α,β)-reverse derivations associated with (α,β)-reverse derivations of *-prime rings is characterized. The action of these derivations on *-Lie ideals of *-prime rings is also investigated. Moreover, the commutativity of *-prime rings admitting multiplicative generalized (α,β)-reverse derivations associated with (α,β)-reverse derivations satisfying certain algebraic identities on *-Lie ideals is explored. [ABSTRACT FROM AUTHOR]

This paper studies commutativity of sixth-order differential systems with zero and nonzero initial conditions (ICs). Given a sixth-order differential system A , we find its commutative pair, that is a new differential system B of order m ≤ 6 ; the obtained result will be used to show the input–output equivalency between the cascaded-connected systems AB and BA . Necessary and sufficient conditions for commutativity of sixth-order continuous-time linear time-varying systems (CTLTVSs) are derived. Based on our findings, we discovered that only a small group of sixth-order CTLTVSs has commutative pairs other than their feedback conjugates obtained by constant feedback and forward path gains. The explicit results obtained in this paper will help to fill the gap on commutativity theory, system behaviors which has significant application to science and play an important role in engineering. This research will also address the problems of sensitivity due to changes in ICs or parameters, and also disturbance as a result of external noise. The results are well verified by examples as well as validated by Matlab Simulink tool and Wolfram Mathematica 11. [ABSTRACT FROM AUTHOR]

This paper considers a problem for Bayesian epistemology and proposes a solution to it. On the traditional Bayesian framework, an agent updates her beliefs by Bayesian conditioning, a rule that tells her how to revise her beliefs whenever she gets evidence that she holds with certainty. In order to extend the framework to a wider range of cases, Jeffrey (1965) proposed a more liberal version of this rule that has Bayesian conditioning as a special case. Jeffrey conditioning is a rule that tells the agent how to revise her beliefs whenever she gets evidence that she holds with any degree of confidence. The problem? While Bayesian conditioning has a foundationalist structure, this foundationalism disappears once we move to Jeffrey conditioning. If Bayesian conditioning is a special case of Jeffrey conditioning, then they should have the same normative structure. The solution? To reinterpret Bayesian updating as a form of diachronic coherentism. [ABSTRACT FROM AUTHOR]

In this paper, we point out some errors in Ali et al. (Bull Iran Math Soc, https://doi.org/10.1007/s41980-019-00231-5, 2019). [ABSTRACT FROM AUTHOR]

In this paper we investigate identities with two generalized derivations in prime rings. Let R be a 2-torsion free prime ring admitting two generalized derivations F and G, not both zero. Among others, we prove that if F(xy) + G(yx) 2 Z(R) for all x; y 2 R, then R is a commutative. Also, if the ring R is equipped with an involution of the second kind and F(xx) + G(x+x) 2 Z(R) for all x 2 R, then R is commutative. The proved theorems give a rise to many corollaries which recover well-known results on (generalized) derivations and left multiplier maps on prime rings (resp. with involution). All along the paper, examples are given to discuss the necessity of our assumptions. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with center Z, I a nonzero ideal of R and D : R x R → R a symmetric bi-(α, α)-derivation and d be the trace of D. In the present paper, we have considered the following conditions: i) [d(x),x]α,α = 0, ii)[d(x), x] α,α ⊆ Cα,α, iii)(d(x), x) α,α = 0, iv)D1(d2(x), x) = 0, v)d1(d2(x)) = f (x), for all x, y ∈ I, where D1 and D2 are two symmetric bi-(α, α)-derivations, d1, d2 are the traces of D1,D2 respectively, B : R x R → R is a symmetric bi-additive mapping, f is the trace of B. [ABSTRACT FROM AUTHOR]

In this paper we consider a more general class of centralizers called I-centralizers. More precisely, given a prime ideal P of an arbitrary ring R we establish a connection between certain algebraic identities involving a pair of P-left centralizers and the structure of the factor ring R/P. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the commutativity of prime rings with involution * of the second kind in which endomorphisms satisfy some algebraic identities involving hermitian and skew-hermitian elements. We will also provide some classifications of these endomorphisms. Finally, we give some examples to prove that the imposed hypotheses are necessary. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we mention some errors in Communications in Algebra, DOI:10.1080/00927872.2019.1684510, and give a correct version of the errant results. In the second part, we generalize some results in Communications in Algebra (DOI:10.1080/00927872.2016.1274900) and present our contribution in the study of generalized two-sided α-derivations satisfying certain identities over near-rings. [ABSTRACT FROM AUTHOR]

In this paper, the notion of generalized derivations is characterized. Also, the action of these derivations on ideals is investigated and as a consequence, some results involving commutator identities are proved. Further, we explore the commutativity of prime additively inverse semirings in which generalized derivations satisfy certain differential identities. An extension to Posner's second theorem is also established in the framework of generalized derivations. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to develop a new approach that consists to investigate the structure of a quotient ring R/P via action of generalized derivations on the prime ideal P. In this direction, we initiate new classes of additive mappings extending both of centralizing and commuting mappings. Furthermore, for an arbitrary ring, we will consider algebraic identities based on its prime ideals. As an application, we obtain some results concerning invariance of minimal prime ideals of a semiprime ring under derivations. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with characteristic different from two and F be a generalized derivation associated with a derivation d of R. In the present paper we investigate the commutativity of R with Jordan involution. Further, examples are given to demonstrate that the restrictions imposed on the hypothesis of our results are not superfluous. [ABSTRACT FROM AUTHOR]

Suppose that N is a near-ring and P is a 3-prime ideal of N. In this paper we introduce the notion of (α, τ)-P derivation in near-rings, we also study the structure of the quotient near-ring N=P which satisfies certain algebraic identities involving (α, τ)-P derivations. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the notion of multiplicative semiderivation and prove some theorems in the setting of a semigroup ideal of a 3-prime near-ring admitting a multiplicative semiderivation. Thereby, it is shown that under appropriate additional hypotheses a near-ring must be a commutative ring. Furthermore, an example is given to illustrate that the 3-primeness hypothesis is not superfluous. [ABSTRACT FROM AUTHOR]

Consider a linear time-varying (LTV) system described by the state-space equation x = A(t)x + B(t)u(t). The main objectives of this paper include (i) determination of the analytical or closed-form solutions for the fundamental matrix X(t) and the state transition matrix (t,to) Φ of the LTV system; (ii) design of feedback control, such that the closed-loop system matrix Acl (t) = A(t)-B(t)K(t), where K(t) is a gain matrix, has desirable properties, in particular, Acl (t) being commutative and triangular; and (iii) design of observers such that the observer matrix Ao(t)=A(t)-H(t)C(t), where H(t) is the observer gain matrix, has desirable properties as in (ii), namely, Ao(t) being commutative and triangular. The commutativity and triangularization of Acl(t) and Ao(t) facilitate the analytical solutions for their fundamental and state transition matrices. Examples and simulations demonstrate the design objectives. [ABSTRACT FROM AUTHOR]

This paper investigates properties of words such that their Parikh q-matrices commute. For any two words u , v ∈ ∑ k * , the concept of q-weak ratio property between u and v is introduced. Based on this property a necessary and sufficient condition for the Parikh q-matrices of any two words over binary alphabet to commute is given. This paper also provides a special class of words over an arbitrary alphabet ∑k such that their Parikh q-matrices commute. [ABSTRACT FROM AUTHOR]

In this paper we give a classification of endomorphisms and additive mappings of a prime ring satisfying certain algebraic identities. Moreover, we provide an example proving that the primeness hypothesis imposed in our theorems is not superfluous. [ABSTRACT FROM AUTHOR]

This paper presents an analysis of a hitherto barely known conceptual problem in the foundations of arithmetic. An unknown type of number, so-called complex numbers, first emerged in 18th century France and became connected with the claim of non-commutativity of their multiplication. These first French practitioners are here analysed, along with examination of the impact of the introduction of the metric decimal system. International dissemination of the notion is then analysed, in particular in the case of Brazil. The paper concludes by looking at the mathematical foundations of multiplying (physical) quantities. [ABSTRACT FROM AUTHOR]

Our goal in the present paper is to study a connection between the commutativity of rings and the behaviour of its generalized derivations. More specifically, we investigate commutativity of quotient rings R/P where R is any ring and P is a prime ideal of R which admits generalized derivations satisfying certain algebraic identities acting on prime ideal P without the primeness (semi-primeness) assumption on the considered ring. This approach allows us to generalize some well known results characterizing commutativity of rings. [ABSTRACT FROM AUTHOR]

Let $ \mathcal{A} $ be a Banach algebra and $ n > 1 $, a fixed integer. The main objective of this paper is to talk about the commutativity of Banach algebras via its projections. Precisely, we prove that if $ \mathcal{A} $ is a prime Banach algebra admitting a continuous projection $ \mathcal{P} $ with image in $ \mathcal{Z}(\mathcal{A}) $ such that $ \mathcal{P}(a^n) = a^n\; \text{for all} \; a \in \mathcal{G} $, the nonvoid open subset of $ \mathcal{A} $, then $ \mathcal{A} $ is commutative and $ \mathcal{P} $ is the identity mapping on $ \mathcal{A} $. Apart from proving some other results, as an application we prove that, a normed algebra is commutative iff the interior of its center is non-empty. Furthermore, we provide some examples to show that the assumed restrictions cannot be relaxed. Finally, we conclude our paper with a direction for further research.

Let R be a prime ring of characteristic not 2 and U be a noncentral square closed Lie ideal of R: An additive mapping H on R is called a homoderivation if H(xy) = H(x)H(y)+H (x) y+xH(y) for all x, € 2 R. In this paper we investigate homoderivations satisfying certain differential identities on square closed Lie ideals of prime rings. [ABSTRACT FROM AUTHOR]