相关论文: The Magnus expansion, trees and Knuth's rotation c…

The notion of trees plays an important role in Butcher's B-series. More recently, a refined understanding of algebraic and combinatorial structures underlying the Magnus expansion has emerged thanks to the use of rooted trees. We follow…

The Magnus expansion, introduced by Wilhelm Magnus in 1954, is an infinite Lie series employed to express solutions for first-order homogeneous linear differential equations involving a linear operator. Since its discovery it has evolved…

We provide a refined approach to the classical Magnus and Fer expansion, unveiling a new structure by using the language of dendriform and pre-Lie algebras. The recursive formula for the logarithm of the solutions of the equations X=1+ta<X…

We address the problem of constructing the non-associative version of the Dynkin form of the Baker-Campbell-Hausdorff formula; that is, expressing $\log (\exp (x)\exp(y))$, where $x$ and $y$ are non-associative variables, in terms of the…

We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part IE, we consider the…

In this paper an application of the recently introduced pre-Lie Magnus expansion to Jackson's q-integral and q-exponentials is presented. Twisted dendriform algebras, which are the natural algebraic framework for Jackson's q-analogues, are…

Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in…

We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part IA, we consider…

We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part II, we consider the…

The Magnus expansion provides an exponential representation of one-parameter operator families, expressed as a series expansion in its generators. This is useful for example in quantum mechanics for expressing a unitary evolution determined…

Let ${\mathbb M}_{g, 1}$, $g \geq 1$, be the moduli space of triples $(C, P_0, v)$ of genus $g$, where $C$ is a compact Riemann surface of genus $g$, $P_0 \in C$, and $v \in T_{P_0}C\setminus\{0\}$. Using Chen's iterated integrals we…

Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to…

Forest formulas that generalize Zimmermann's forest formula in quantum field theory have been obtained for the computation of the antipode in the dual of enveloping algebras of pre-Lie algebras. In this work, largely motivated by Murua's…

In this report the emphasis is on an alternative representation of the Magnus series by proper operator (matrix) exponential solutions to differential equations (systems), both linear and nonlinear ODEs and PDEs. The main idea here is in…

We review the discrete evolution problem and the corresponding solution as a discrete Dyson series in order to rigorously derive a generalized discrete version of the Magnus expansion. We also systematically derive the discrete analogue of…

We present a general expression for any term of the Magnus series as an iterated integral of a linear combination of independent right-nested commutators with given coefficients. The relation with the Malvenuto--Reutenauer Hopf algebra of…

We identify the Baker-Campbell-Hausdorff recursion driven by a weight$\lambda=1$ Rota-Baxter operator with the Magnus expansion relativeto the post-Lie structure naturally associated to the correspondingRota-Baxter algebra. Post-Lie Magnus…

Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of…

The Magnus expansion offers a method to express a time-ordered exponential as an ordinary operatorial exponential. This representation has advantageous theoretical properties, while still solving the original differential equation. For any…

In this paper, we study the classical and pre-Lie Magnus expansions, discussing how we can find a recursion for the pre-Lie case which already incorporates the pre-Lie identity. We give a combinatorial vision of a numerical method proposed…