Narrow Results

In this work, we propose a definition of comonotonicity for elements of $B{(H)}_{sa}$, i.e. bounded self-adjoint operators defined over a complex Hilbert space $H$. We show that this notion of comonotonicity coincides with a form of commutativity. Intuitively, comonotonicity is to commutativity as monotonicity is to bounded variation. We also define a notion of Choquet expectation for elements of $B{(H)}_{sa}$ that generalizes quantum expectations. We characterize Choquet expectations as the real-valued functionals over $B{(H)}_{sa}$ which are comonotonic additive, $c$-monotone, and normalized.

On the basis of the generalization of the theorem about K-odd operators (K is the Dirac's operator), certain linear combination is constructed, which appears to commute with the Dirac Hamiltonian for Coulomb field. This operator coincides with the Johnson and Lippmann operator and is closely connected to the familiar Laplace–Runge–Lenz vector. Our approach guarantees not only derivation of Johnson–Lippmann operator, but simultaneously commutativity with the Dirac Hamiltonian.

In this work, we present a comprehensive study of the commutativity of fractional-order linear time-varying systems (LTVSs). Commutativity is a fundamental property in the analysis and control of dynamic systems and is often used to simplify the design of controllers. Fractional-order systems, which are characterized by a noninteger-order derivative, have been widely studied in recent years due to their ability to model a wide range of phenomena. However, the commutativity of fractional-order LTVSs has not been widely explored. In this work, we present a comprehensive study of the commutativity of fractional-order LTVSs. We first provide a mathematical definition of commutativity for these systems and demonstrate that it is equivalent to the commutativity of their transfer functions. We then propose a method for verifying the general condition for commutativity of fractional-order LTVSs under zero initial conditions (ICs) and prove it mathematically. Based on our findings, we realized that the commutative requirements, properties, theories, and conditions are general for fractional-order LTVSs, please observed that some fractional-order LTVSs are commutative, some are not commutative, while some are commutative under certain conditions. Based on this fact, we can say that not all fractional-order LTVSs are commutative.

We apply explicit commutative results to several examples of fractional-order LTVSs. Our theoretical and simulation results show a good agreement and prove that our fractional-order LTVSs are commutative under certain conditions, moreover, the commutativity property holds for certain conditions and classes of fractional-order LTVSs, but not for others. Because of the application of fraction commutativity in various fields of science and engineering, we find it necessary to come up with explicit results for the first time.

We show that, when restricted to the class of varieties that have a Taylor term, several commutator properties are definable by Maltsev conditions.

Let R be an associative ring and let x, y ∈ R. Define the generalized commutators as follows: [x, ₀y] = x and [x, _ky] = [x, _k-1y]y - y[x, _k-1y](k = 1, 2, …). In this paper we study some generalized Engel rings, i.e. -rings (satisfying [x^{m(x, y)}, _{k(x, y)}y] = 0), -rings (satisfying [x^{m(x, y)}, _{k(x, y)}y^{n(x, y)}] = 0) and -rings (satisfying [x^{m(x, y)}, _{k(x, y)}y^{n(x, y)}]^{r(x, y)} = 0). Among other results, it is proved that every Artinian -ring is strictly Lie-nilpotent. Also, we show that in each of the following cases R has nil commutator ideal: (1) if R is a -ring with unity and k, n independent of y; (2) if R is a locally bounded -ring (defined below); (3) if R is an algebraic algebra over a field in which R* is a bounded Engel group or a soluble group.

In this paper, we investigate commutativity of prime rings $R$ with involution ∗ of the second kind in which generalized derivations satisfy certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Furthermore, we provide an example to show that the restriction imposed on the involution is not superfluous.

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 ${𝒜}$ admitting a derivation satisfying certain algebraic identities.

In this paper, we introduce dual g-frames for a closed subspace in a separable Hilbert space and also give a characterization. Generally dual g-frames for a closed subspace are non-commutative. Therefore, we construct dual g-frames for closed subspaces from two aspects and give the corresponding formulas, respectively. Finally, we give a necessary and sufficient condition for commutative dual g-frame pairs for closed subspaces under certain conditions.

In this paper we investigate commutativity of prime rings with involution ∗ of the second kind in which endomorphisms satisfy certain algebraic identities. Furthermore, we provide examples to show that the various restrictions imposed by the hypotheses of our theorems are not superfluous.

In this paper it is shown that zero symmetric prime left near-rings satisfying certain identities involving semiderivations are commutative rings.

In this paper, we investigate generalized derivations satisfying certain differential identities on semigroup ideals of right near-rings and discuss related results. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

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. The purpose of this paper is to study generalized semiderivations which satisfy certain identities on 3-prime near-ring and generalize some results due to [H. E. Bell and G. Mason, On derivations in near-rings, North-Holland Math. Stud.137 (1987) 31–35; H. E. Bell, On prime near-rings with generalized derivation, Int. J. Math. Math. Sci.2008 (2008), Article ID: 490316, 5pp; A. Boua and L. Oukhtite, Some conditions under which near-rings are rings, Southeast Asian Bull. Math.37 (2013) 325–331]. Moreover, an example is given to prove that the necessity of the 3-primeness hypothesis imposed on the various theorems cannot be marginalized.

We prove commutativity of rings $R$ with 1 which satisfy certain $n$th power identities on proper subsets of $R$.

Our purpose in this paper is to investigate commutativity of a ring with involution $(R,\ast )$ which admits a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

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.

The main purpose of this paper is to study certain central-valued identities on a factor ring with respect to a prime ideal ${𝒫}$ 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.

For fixed integers $1\le i<j\le n$ and $\sigma \in S_n$, we study the faithful representation of superassociative algebras by two kinds of functions of the arity $n$, the so called $(i,j)$-commutative and $\sigma$-commutative. The automorphism on the algebra of all $(i,j)$-commutative functions is determined. In general, an isomorphism from a superassociative system which consists of a family of nonempty sets and a family of operations satisfying the axiom of superassociativity to a system of full multiplace functions is discussed. Particularly, we also provide necessary and sufficient conditions under which a superassociative system and a system of $\sigma$-commutative functions are isomorphic. With these results, a close connection with the theory of terms and tree languages is described.

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.

The overall goal of this research is to explore the interactions that exist between the structure of a quotient ring $R/P$ and the behavior of generalized derivations of an arbitrary ring with involution $R$. To be more exact, we use two generalized derivations of $R$ that satisfy algebraic identities involving the prime ideal $P$ to describe the commutativity of $R/P$. Besides, we offer instances to demonstrate that the multiple constraints necessary in the assumptions of our theorems are not excessive.

Let $R$ be a ring and $P$ a prime ideal of $R.$ This paper establishes a connection between the commutativity of the factor ring $R/P$ and semiderivations of $R$ satisfying some algebraic identities involving the prime ideal $P.$ Moreover, we extended some results for derivations and semiderivations of prime rings to semiprime rings.