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In this work, we define the class preserving actor and commutativity degree of Lie crossed modules. Then, we obtain some relations about these notions and isoclinic Lie crossed modules. In this work, we define the class preserving actor and commutativity degree of Lie crossed modules. Then, we obtain some relations about these notions and isoclinic Lie crossed modules.

For factorizing representative (or rational) series, with coefficients in a commutative ring A containing ℚ, we examine various products such as concatenation, shuffle and its ϕ -deformations, … (and their co-products) defined on the free monoid which are such that their associated bialgebras are isomorphic to the Sweedler’s dual, for A being a field K .

We discuss some features of thermodynamics in the presence of multiple conserved quantities. We prove a generalisation of Landauer principle illustrating tradeoffs between the erasure costs paid in different 'currencies'. We then show how the maximum entropy and complete passivity approaches give different answers in the presence of multiple observables. We discuss how this seems to prevent current resource theories from fully capturing thermodynamic aspects of non-commutativity.

Let A be a prime ring, Z(A) its center, Q its right Martindale quotient ring, C its extended centroid, ψ a non-zero b-generalized derivation of A with associated map ξ. In this article, we prove that: (i) If [ψ(x), ψ(y)] = 0 for all x, y ∈ A, then A is either commutative or there exists q ∈ Q such that ξ = ad(q), ψ(x) = -bxq, and qb = 0. (ii) If ψ(x) ◦ ψ(y) = 0 for all x, y ∈ A, then A is either commutative with char(A) = 2 or there exists q ∈ Q such that ψ(x) = -bxq and qb = 0. Additional results are established for cases involving [ξ(x), ψ(x)] = 0 or ξ(x)◦ψ(x) = 0, where char(A) = 2. Furthermore, we give some examples that show the importance of the hypotheses of our theorems. Sea A un anillo primo, Z(A) su centro, Q su anillo de cocientes de Martindale por derecha, C su centroide extendido, ψ una derivada b-generalizada de A con mapa asociado ξ. En este artículo probamos los siguientes resultados: (i) Si [ψ(x), ψ(y)] = 0 para todo x, y ∈ A, entonces o A es conmutativo o existe q ∈ Q tal que ξ = ad(q), ψ(x) = -bxq, y qb = 0. (ii) Si ψ(x) ◦ ψ(y) = 0 para todo x, y ∈ A, entonces o A es conmutativo con char(A) = 2 o existe q ∈ Q tal que ψ(x) = -bxq y qb = 0. También se analizan los casos donde [ξ(x), ψ(x)] = 0 o ξ(x) ◦ ψ(x) = 0, donde char(A) = 2. Se incluyen ejemplos que ilustran la importancia de las hipótesis de los teoremas.

This study aims to investigate the commutativity of a prime ring R with a non-zero ideal I and a homo-derivation ů that satisfies certain algebraic identities. We also provided some examples of why our results hypothesis is essential.

The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds, in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure,k-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a by-product. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of 5, and it is close to tight for power-law interactions and other orderings of terms. This result suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.

Let be a Banach algebra. In this article, on the one hand, we proved some results concerning the continuous projection from to its center. On the other hand, we investigate the commutativity of under specific conditions. Finally, we included some examples and applications to prove that various restrictions in the hypotheses of our theorems are necessary.

We present the notion of generalized reverse derivation on Γ-semiring in this article and we prove that a prime Γ-semiring with center CΓ stratifying assumption (*) a α b β c = a β b α c, for all a, b, c S and α , Γ and I be a non-zero right ideal of S such that S admits a generalized reverse derivation f associated with a reverse derivation d satisfying d(CΓ) ≠ 0, if S satisfies any one of the properties: (i) f(aαb) + aαb Γ (ii) f([a, b]α) + [f(a), b]α Γ (iii) f([a, b]α) + [f(a), f(b)]α Γ (iv) [f(a), b]α + [a, f(b)]α Γ For of the all a, b I and α Γ, then S is commutative.

The concept of Γ-semiring is a generalization of semiring. Two important mappings of Γ-semirings are derivations and generalized derivations. The commutativity of prime Γ-semiring with derivation and generalized derivation f associated with the d derivation is discussed in this article.

Let R be a nonassociative ring with center U. In this paper, it is shown that nonassociative ring R of char. ≠ 2 with unity is commutative if it satisfies any one of the following identities: (i) (xy)x + x(xy) + y∈ U, (ii) (xy)2 - x2 y - xy2 - xy∈ U, (iii) (xy)2 - x2 y - xy2 - yx∈ U (iv) (xy)2- xy2∈ U, (v) (xy)2- y2 x ∈ U, (vi) (x2y2)z2 - (xy)z ∈ U, (vii) (x2y2)z2-(xy)z ∈ U for all x, y, and for fixed z in R.