A general formula for the Magnus expansion in terms of iterated integrals of right-nested commutators

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 permutations is also discussed.Comment: 16 page

La evolución del arte fantástico en México : Daniel Lezama

Es importante para la historia del arte mexicano considerar la posibilidad de una continuidad del arte fantástico, desde José Guadalupe Posada hasta la pintura figurativa de fin del siglo XX y comienzos de XXI. Mi propuesta se basa en el libro de la historiadora Ida Rodríguez Prampolini, El surrealismo y el arte fantástico de México, escrito en 1969. Para una definición más amplia e incluyente recurro a la tesis literaria del escritor mexicano Omar Nieto Arroyo, publicada en 2015. Desde esta lejanía podremos ver claramente el hilo conductor de la producción fantástica mexicana, y por ende lo más importante: la estupenda maleabilidad del género para mostrar todo tipo de conceptos. Como un ejemplo del arte fantástico de México en la actualidad presento la obra del pintor Daniel Lezama.It is important for the history of Mexican art to consider the possibility of a continuity in fantastic art, from José Guadalupe Posada to the figurative painting of the end of the twentieth and beginning of the twenty-first century. My proposal is based on the historian Ida Rodríguez Prampolini's book El surrealismo y el arte fantástico de México [Surrealism and fantastic art in Mexico], written in 1969. For a broader and more inclusive definition I make use of Omar Nieto Arroyo's literary thesis published in 2015. From this distance we will be able to see clearly the thread unifying Mexican production of the fantastic, and thus what is most important: the wonderful malleability of the genre to show all manner of conceits. I use Daniel Lezama as an example of contemporary Mexican art

A Lie-Deprit perturbation algorithm for linear differential equations with periodic coefficients

A perturbative procedure based on the Lie-Deprit algorithm of classical mechanics is proposed to compute analytic approximations to the fundamental matrix of linear di erential equations with periodic coe cients. These approximations reproduce the structure assured by the Floquet theorem. Alternatively, the algorithm provides explicit approximations to the Lyapunov transformation reducing the original periodic problem to an autonomous sys- tem and also to its characteristic exponents. The procedure is computationally well adapted and converges for su ciently small values of the perturbation pa- rameter. Moreover, when the system evolves in a Lie group, the approximations also belong to the same Lie group, thus preserving qualitative properties of the exact solution

A note on the Baker–Campbell–Hausdorff series in terms of right-nested commutators

We get compact expressions for the Baker–Campbell–Hausdorff series Z = log(eX eY ) in terms of right-nested commutators. The reduction in the number of terms originates from two facts: (i) we use as a starting point an explicit expression directly involving independent commutators and (ii) we derive a complete set of identities arising among right-nested commutators. The procedure allows us to obtain the series with fewer terms than when expressed in the classical Hall basis at least up to terms of grade 10

Exponential Perturbative Expansions and Coordinate Transformations

We propose a unified approach for different exponential perturbation techniques used in the treatment of time-dependent quantum mechanical problems, namely the Magnus expansion, the Floquet–Magnus expansion for periodic systems, the quantum averaging technique, and the Lie–Deprit perturbative algorithms. Even the standard perturbation theory fits in this framework. The approach is based on carrying out an appropriate change of coordinates (or picture) in each case, and it can be formulated for any time-dependent linear system of ordinary differential equations. All of the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation

A unifying framework for perturbative exponential factorizations

We propose a framework where Fer and Wilcox expansions for the solution of differential equations are derived from two particular choices for the initial transformation that seeds the product expansion. In this scheme intermediate expansions can also be envisaged. Recurrence formulas are developed. A new lower bound for the convergence of the Wilcox expansion is provided as well as some applications of the results. In particular, two examples are worked out up to high order of approximation to illustrate the behavior of the Wilcox expansion

Exponential perturbative expansions and coordinate transformations

We propose a unified approach for different exponential perturbation techniques used in the treatment of time-dependent quantum mechanical problems, namely the Magnus expansion, the Floquet--Magnus expansion for periodic systems, the quantum averaging technique and the Lie--Deprit perturbative algorithms. Even the standard perturbation theory fits in this framework. The approach is based on carrying out an appropriate change of coordinates (or picture) in each case, and can be formulated for any time-dependent linear system of ordinary differential equations. All the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schr\"odinger equation

Magnus integrators for linear and quasilinear delay differential equations

A procedure to numerically integrate non-autonomous linear delay differential equations is presented. It is based on the use of an spectral discretization of the delayed part to transform the original problem into a matrix linear ordinary differential equation which is subsequently solved with numerical integrators obtained from the Magnus expansion. The algorithm can be used in the periodic case to get both accurate approximations of the characteristic multipliers and the solution itself. In addition, it can be extended to deal with certain quasilinear delay equations