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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.

The objective of this paper is to generalize the concept of commutativity degree of a finite polygroup P , by defining the notion of relative commutativity degree of a subpolygroup S of a polygroup P . In this regard, we state some results concerning the newly defined concept. Then, we illustrate through examples that the known results for groups may not hold for polygroups. Finally, we compute the relative commutativity degree of subpolygroups of some special polygroups.

The aim of this paper is to determine the possible values of the commutativity degree of Lie algebras. We define the asymptotic commutativity degree of Lie algebras and obtain the asymptotic commutativity degree for some of them. Moreover, we prove the existence of a family of Lie algebras such that the asymptotic commutativity degree is equal to 1/ q k for all q ≥ 2 and a positive integer k .

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.

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.

We extend the theory of ambidexterity developed by M.J. Hopkins and J. Lurie by proving commutativity of the norm square induced from a weakly ambidextrous morphism by two Beck-Chevalley fibrations that are associated by a functor. By showing how ambidexterity is preserved under base change of Beck-Chevalley fibrations, we demonstrate that our result is a generalization of the naturality property of the norm shown by M.J. Hopkins and J. Lurie. Furthermore, we demonstrate how our generalization implies two specific results previously shown by S. Carmeli, T. M. Schlank, and L. Yanovski, namely, that the induced norm square of local systems, and the induced norm square of equivariant powers, both commute.

This article is written in celebration of the 8th Kazakh-French Logical Colloquium. We expand on an unpublished research note of the second author. We record some results concerning local Keisler measures with respect to a formula which is stable in a model. We prove that in this context, every local Keisler measure on the associated local type space is a weighted sum of (at most countably many) types. Using this observation, we give an elementary proof of the commutativity of the Morley product in this context. We then give a functional analytic proof that the double limit property lifts to the appropriate evaluation map on pairs of local measures. We end with some comments on the NOP and local measures in the (properly) stable context.

We continue the study of the pseudo-intersection property with respect to an ideal introduced in TomNatasha2. Our theory applies to the study of the Tukey types of general sums of ultrafilters, which, as evidenced by the results of this paper, can be quite complex. It also applies to construct a large class of ultrafilter \\(C\\) over \\(\\) such that any two ultrafilters \\(U,Vın C\\) commute; that is, \\(U V_T V U\\). The class \\(C\\) class contains most known cofinal types of ultrafilters on \\(\\). This is in sharp contrast to the Rudin-Keisler ordering. In the third part of this paper, we apply our results to study the class of ultrafilters Tukey above \\(^\\). Specifically, we prove that ultrafilters without the \\(I\\)-p.i.p are always above \\(I^\\) and in particular non-\\(p\\)-points are Tukey above \\(^\\). Finally, we introduce the hierarchy of \\(\\)-almost rapid ultrafilters. We prove that it is consistent for them to form a strictly wider class than the rapid ultrafilters, and give an example of a non-rapid \\(p\\)-point ultrafilter which is Tukey above \\(^\\). This addresses and answers several questions from TomNatasha,TomNatasha2,Dobrinen/Todorcevic11,Milovich08.