Twistability of Spin Particle Trajectories

Using the geometric optics approximation, we establish the existence of the additional twisting effect for trajectories of spin particles, i.e., the analogue of the Magnus optical effect. The effect is determined by the polarization (chirality) and the curvature of the particle trajectory. We investigate the proton scattering in the Coulomb field of a nucleus and the mirror reflection law breaking for ultracold neutrons.     

Polarization effects due to the mutual influence of trajectory parameters and polarization

In the geometric optics approximation, we comparatively analyze the polarization effects resulting from the influence of polarization on the beam trajectory. We show that the beam trajectory equations describing the optical Magnus effect that are obtained from the canonical Hamilton equations, Fermat principle, and truncated vector wave equations give the same result, coincident with the result in the mode approach. We explain the reasons underlying the previously derived results. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 65–79, October, 2006.     

Additional Curvature of Spinning-Particle Trajectories Proportional to the Trajectory Torsion

In the geometric optics approximation, we predict the effect of an additional curvature of the trajectories of particles (an analogue of the inverse Magnus optical effect). The effect is determined by the polarization (chirality) and torsion of particle trajectories. We find that the considered effect is linked to the Berry phase. The effect is a consequence of the conservation law for the angular momentum of the particles. We show that the effect must result in ultracold neutrons deviating from the mirror reflection law.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 555–563, September, 2005.     

An Analogue of the Magnus Problem for Associative Algebras

We prove an analogue of the Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by n + k generators and k relations and has an n-element system of generators, then this algebra is a free algebra of rank n. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.     

The Magnus Embedding for Right-Symmetric Algebras

We construct a basis for the universal multiplicative enveloping algebra U(A) of a right-symmetric algebra A. We prove an analog of the Magnus embedding for right-symmetric algebras; i.e., we prove that a right-symmetric algebra A/R ², where A is a free right-symmetric algebra, is embedded into the algebra of triangular matrices of the second order.     

Numerical Evaluation of the Evans Function by Magnus Integration

We use Magnus methods to compute the Evans function for spectral problems as arise when determining the linear stability of travelling wave solutions to reaction-diffusion and related partial differential equations. In a typical application scenario, we need to repeatedly sample the solution to a system of linear non-autonomous ordinary differential equations for different values of one or more parameters as we detect and locate the zeros of the Evans function in the right half of the complex plane. In this situation, a substantial portion of the computational effort—the numerical evaluation of the iterated integrals which appear in the Magnus series—can be performed independent of the parameters and hence needs to be done only once. More importantly, for any given tolerance Magnus integrators possess lower bounds on the step size which are uniform across large regions of parameter space and which can be estimated a priori. We demonstrate, analytically as well as through numerical experiment, that these features render Magnus integrators extremely robust and, depending on the regime of interest, efficient in comparison with standard ODE solvers. AMS subject classification (2000) 65F20     

Quasi-hopf algebra actions and smash products

A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group F^A has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that F^A is faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and F^A is a free A-groupp of Lyndon' type, then F^A has a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then F^A has a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, F^A has a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if F^Q has a faithful Magnus representation in Q[[Y]].     

On the Global Error of Discretization Methods for Highly-Oscillatory Ordinary Differential Equations

Commencing from a global-error formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highly-oscillating systems of the form y+ g(t)y = 0, where g(t) . Using WKB analysis we derive an explicit form of the global-error envelope for Runge-Kutta and Magnus methods. Our results are closely matched by numerical experiments.Motivated by the superior performance of Lie-group methods, we present a modification of the Magnus expansion which displays even better long-term behaviour in the presence of oscillations.     

Residual properties of free groups,III

In this paper we want to prove the following theorem: Letx be an infinite set of non-abelian finite simple groups. Then the free groupF ₂ on 2 generators is residuallyx. This answers a question first posed by W. Magnus and later by A. Lubotzky [9], Yu. Gorchakov and V. Levchuk [4]. The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft.     

On the sensitivity of the one-sided test to covariance misspecification

Sensitivity analysis stands in contrast to diagnostic testing in that sensitivity analysis aims to answer the question of whether it matters that a nuisance parameter is non-zero, whereas a diagnostic test ascertains explicitly if the nuisance parameter is different from zero. In this paper, we introduce and derive the finite sample properties of a sensitivity statistic measuring the sensitivity of the t statistic to covariance misspecification. Unlike the earlier work by Banerjee and Magnus [A. Banerjee, J.R. Magnus, On the sensitivity of the usual t- and F-tests to covariance misspecification, Journal of Econometrics 95 (2000) 157–176] on the sensitivity of the F statistic, the theorems derived in the current paper hold under both the null and alternative hypotheses. Also, in contrast to Banerjee and Magnus’ [see the above cited reference] results on the F test, we find that the decision to accept the null using the OLS based one-sided t test is not necessarily robust against covariance misspecification and depends much on the underlying data matrix. Our results also indicate that autocorrelation does not necessarily weaken the power of the OLS based t test.     

Solving ODEs arising from non‐selfadjoint Hamiltonian eigenproblems

We consider three numerical methods – one based on power series, one on the Magnus series and matrix exponentials, and one a library initial value code – for solving a linear system arising in non‐selfadjoint ODE eigenproblems. We show that in general, none of these methods has a cost or an accuracy which is uniform in the eigenparameter, but that for certain special types of problem, the Magnus method does yield eigenparameter‐uniform accuracy. This property of the Magnus method is explained by a trajectory‐shadowing result which, unfortunately, does not generalize to higher order Magnus type methods such as those in [11,12]. This revised version was published online in June 2006 with corrections to the Cover Date.     

Magnus’ expansion for time-periodic systems: Parameter-dependent approximations

Magnus’ expansion solves the nonlinear Hausdorff equation associated with a linear time-varying system of ordinary differential equations by forming the matrix exponential of a series of integrated commutators of the matrix-valued coefficient. Instead of expanding the fundamental solution itself, that is, the logarithm is expanded. Within some finite interval in the time variable, such an expansion converges faster than direct methods like Picard iteration and it preserves symmetries of the ODE system, if present. For time-periodic systems, Magnus expansion, in some cases, allows one to symbolically approximate the logarithm of the Floquet transition matrix (monodromy matrix) in terms of parameters. Although it has been successfully used as a numerical tool, this use of the Magnus expansion is new. Here we use a version of Magnus’ expansion due to Iserles [Iserles A. Expansions that grow on trees. Not Am Math Soc 2002;49:430–40], who reordered the terms of Magnus’ expansion for more efficient computation. Though much about the convergence of the Magnus expansion is not known, we explore the convergence of the expansion and apply known convergence estimates. We discuss the possible benefits to using it for time-periodic systems, and we demonstrate the expansion on several examples of periodic systems through the use of a computer algebra system, showing how the convergence depends on parameters.     

On the Implementation of the Method of Magnus Series for Linear Differential Equations

The method of Magnus series has recently been analysed by Iserles and Nørsett. It approximates the solution of linear differential equations y = a(t)y in the form y(t) = e ^(t) y ₀, solving a nonlinear differential equation for by means of an expansion in iterated integrals of commutators. An appealing feature of the method is that, whenever the exact solution evolves in a Lie group, so does the numerical solution.The subject matter of the present paper is practical implementation of the method of Magnus series. We commence by briefly reviewing the method and highlighting its connection with graph theory. This is followed by the derivation of error estimates, a task greatly assisted by the graph-theoretical connection. These error estimates have been incorporated into a variable-step fourth-order code. The concluding section of the paper is devoted to a number of computer experiments that highlight the promise of the proposed approach even in the absence of a Lie-group structure.     

Nonlinearity effects in multidimensional scaling

When multidimensional scaling of n cases is derived from dissimilarities that are functions of p basic continuous variables, the question arises of how to relate the values of the variables to the configuration of n points. We provide a methodology based on nonlinear biplots that expresses nonlinearity in two ways: (i) each variable is represented by a nonlinear trajectory and (ii) each trajectory is calibrated by an irregular scale. Methods for computing, calibrating and interpreting these trajectories are given and exemplified. Not only are the tools of immediate practical utility but the methodology established assists in a critical appraisal of the consequences of using nonlinear measures in a variety of multidimensional scaling methods.     

Residual properties of free groups,II

In this paper we prove that if X is an infinite class of flnite simple classical groups, then F₂, the free group of rank 2, is residually X. This solves a special case of a question of W.Magnus. He conjectures that F₂ is residually X for any infinite class X of finite non-abelian simple groups.     

复杂地形情况下大气动态模型的弱解和吸引子的存在性

该文研究了一个适合应用于天气预报的大气模型,模型考虑到地形对大气的动力强迫作用,保留了大气气流的辐散效应.首先采用了适当的函数空间,引入了合理的算子表达形式,将复杂的大气动态方程组用一个简单的抽象的算子方程表示,由此给出了模型弱解的定义.然后利用Galerkin方法证明了弱解的存在性.通过构造大气动态方程组轨道吸引集证明相应的轨道吸引子的存在性.     

On the McCool group M3 and its associated Lie algebra

We prove that the Lie algebra of the McCool group M₃ is torsion free. As a result, we are able to give a presentation for the Lie algebra of M₃. Furthermore, M₃ is a Magnus group.     

The Magnus Expansion,Trees and Knuth’s Rotation Correspondence

W. Magnus introduced a particular differential equation characterizing the logarithm of the solution of linear initial value problems for linear operators. The recursive solution of this differential equation leads to a peculiar Lie series, which is known as Magnus expansion, and involves Bernoulli numbers, iterated Lie brackets and integrals. This paper aims at obtaining further insights into the fine structure of the Magnus expansion. By using basic combinatorics on planar rooted trees we prove a closed formula for the Magnus expansion in the context of free dendriform algebra. From this, by using a well-known dendriform algebra structure on the vector space generated by the disjoint union of the symmetric groups, we derive the Mielnik–Plebański–Strichartz formula for the continuous Baker–Campbell–Hausdorff series.     

Book Vignettes

Andrews, Michael, The Life That Lives on Man. Pyke, Magnus, Butter Side Up! The Delights of Science. Stafford, Donald G., Renner, John W., and Rusch, John J., The Physical Sciences: Inquiry and Investigation. Polis, A. Richard and Beard, Earl M. L., Fundamentals of Mathematics: A Cultural Approach.     

Designable Integrability of the Variable Coefficient Nonlinear Schrödinger Equations

The designable integrability (DI) [ 51 ] of the variable coefficient nonlinear Schrödinger equations (VCNLSEs) is first introduced by construction of an explicit transformation, which maps VCNLSE to the usual nonlinear Schrödinger equations (NLSEs). One novel feature of VCNLSE with DI is that its coefficients can be designed artificially and analytically by using transformation. A special example between nonautonomous NLSEs and NLSEs is given here. Further, the optical super‐lattice potentials (or periodic potentials) and multiwell potentials are designed, which are two kinds of important potential in Bose–Einstein condensation and nonlinear optical systems. There are two interesting features of the soliton of the VCNLSEs indicated by the analytic and exact formula. Specifically, its profile is variable and its trajectory is not a straight line when it evolves with time t.