Dynamic reflection and refraction of neutrons

Optical phenomena that arise in the interaction of a neutron wave with matter characterized by a variable magnetic-induction vector are considered. Since the interaction is time-dependent, the neutron energy is not conserved. Only the neutron velocity component, which is orthogonal to the matter boundary, changes. The changes in the neutron energy and in direction of propagation are closely related to the reversal of the neutron-spin projection. Possibilities of application of the thin slab with rotating magnetization are considered.     

Fibrillation of solar magnetic fields

Solar magnetic structures are often observed in the form of flux tubes composed of a number of smaller elements called fibres or threads, although theoretically such concentrations should not appear but should be flattened by magnetic diffusivity into a uniform, low intensity field. In this paper we describe a mechanism which may be responsible for the fibrillation and also for the very large diffusivity which dissipates magnetic flux tubes in hours instead of years. Firstly, the electric current associated with magnetic field gradients usually increases the local electron temperature and reduces the resistivity, so that the current becomes concentrated into sheets or streamers. Secondly, the magnetic field gradients continue to increase until the current magnitude reaches its limit, which is determined by the electron-ion streaming instability. Then with appropriate temperature and number densities, the Larmor radius of the ions overlaps the near discontinuity in B_z and generates a sharply peaked fluid motion at the edge that is close to the thermal speed. Finally, the resulting vorticity generates an axial magnetic field opposing B_z in the term , and if this is sufficient to change the sign of this term, the very unstable backward heat equation results. This instability repeatedly switches on and off and maintains the magnetic structure in the fibrillated form. Such structures are eventually eliminated by magnetic diffusivity in the usual way, but because of the fluctuations in B_z, this occurs at a vastly increased rate. We show that this phenomenon increases the magnetic diffusivity, D, by a factor ~ 10⁸ in agreement with some observations of plasma loops and supergranules.     

1‐Phenyl‐1,2‐cyclohexadiene: Astoundingly High Enantioselectivities on Generation in a Doering–Moore–Skattebøl Reaction and Interception by Activated Olefins

The resolution of (1α,5α,6α)‐6‐bromo‐6‐fluoro‐1‐phenylbicyclo[3.1.0]hexane (rac‐ 5) provided the enantiomerically pure precursors (?)‐ 5 and (+)‐ 5 of 1‐phenyl‐1,2‐cyclohexadiene. On treatment of (?)‐ 5 with methyllithium in the presence of 2,5‐dimethylfuran, the pure (?)‐enantiomer of the [4+2] cycloadduct of 2,5‐dimethylfuran onto 1‐phenyl‐1,2‐cyclohexadiene was obtained exclusively. From this result, it is concluded that pure (M)‐1‐phenyl‐1,2‐cyclohexadiene ((M)‐ 7 ) emerged from (?)‐ 5 and was enantiospecifically intercepted to give the product. In the case of indene as trap for (M)‐ 7 , the (?)‐ and the (+)‐enantiomer of the [2+2] cycloadduct were formed in the ratio of 95:5. Highly surprising, remarkable enantioselectivities were also observed, when (M)‐ 7 was trapped with styrene to furnish two diastereomeric [2+2] cycloadducts. Hence, the achiral conformation of the diradical conceivable as intermediate cannot play a decisive part. The enantioselective generation of (M)‐ and (P)‐ 7 by the β‐elimination route was tested as well. Accordingly, 1‐bromo‐2‐phenylcyclohexene was exposed to the potassium salt of (?)‐menthol in the presence of 2,5‐dimethylfuran, and the enantiomeric [4+2] cycloadducts of the latter onto (M)‐ and (P)‐ 7 were produced in the ratio of 55:45.     

Oxidation and chlorination reactions of perfluoroketene-N,S-acetals

The paper presents the results of the investigation of oxidation and chlorination reactions of perfluoroketene-N,S-acetals. Oxidation reactions of perfluoroketene-N,S-acetals proved to be dependent on the nature of oxidizing agent and led to the formation of corresponding sulfone in the case of m-chloroperbenzoic acid or amides of α-H-perfluoroalkane carboxylic acids in the case of tert-butyl hydroperoxide or hydrogen peroxide. Reaction of 1-tert-butylsulfanyl-2,3,3,4,4,4-hexafluoro-1-[N-methyl,N-((S)-α-methylbenzyl)amino]-but-1-ene with sulfuryl chloride demonstrated the chlorination of perfluoroketene-N,S-acetals as a new approach in the synthesis of chiral α-chloro perfluoroalkane carboxylic acid amides.     

Carleman Estimates for Parabolic Equations with Nonhomogeneous Boundary Conditions

The authors prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On the basis of this estimate, improved Carleman estimates for the Stokes system and for a system of parabolic equations with a penalty term are obtained. This system can be viewed as an approximation of the Stokes system.     

On the solution of the non-local parabolic partial differential equations via radial basis functions

In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions is considered. The approximate solution is found using the radial basis functions collocation method. There are some difficulties in computing the solution of the time dependent partial differential equations using radial basis functions. If time and space are discretized using radial basis functions, the resulted coefficient matrix will be very ill-conditioned and so the corresponding linear system cannot be solved easily. As an alternative method for solution, we can use finite-difference methods for discretization of time and radial basis functions for discretization of space. Although this method is easy to use but an accurate solution cannot be provided. In this work an efficient collocation method is proposed for solving non-local parabolic partial differential equations using radial basis functions. Numerical results are presented and are compared with some existing methods.     

Smoothing schemes for reaction-diffusion systems with nonsmooth data

Cox and Matthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002) 430–455] developed a class of Exponential Time Differencing Runge–Kutta schemes (ETDRK) for nonlinear parabolic equations; Kassam and Trefethen [A.K. Kassam, Ll. N. Trefethen, Fourth-order time stepping for stiff pdes, SIAM J. Sci. Comput. 26 (2005) 1214–1233] have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth-order (ETDRK4) scheme. They use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials, but this approach creates new difficulties in choosing and evaluating the contour for larger problems. Neither treatment addresses problems with nonsmooth data, where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully. Such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications, including computational chemistry and financial engineering. We introduce a new version of the fourth-order Cox–Matthews, Kassam–Trefethen ETDRK4 scheme designed to eliminate the remaining computational difficulties. This new scheme utilizes an exponential time differencing Runge–Kutta ETDRK scheme using a diagonal Padé approximation of matrix exponential functions, while to deal with the problem of nonsmooth data we use several steps of an ETDRK scheme using a sub-diagonal Padé formula. The new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices, having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel. In this approach it is only required that several backward Euler linear problems be solved, in serial or parallel. Numerical experiments are described to support the new scheme.     

The principal eigenvalue of a space–time periodic parabolic operator

This paper deals with the generalized principal eigenvalue of the parabolic operator , where the coefficients are periodic in t and x. We give the definition of this eigenvalue and we prove that it can be approximated by a sequence of principal eigenvalues associated to the same operator in a bounded domain, with periodicity in time and Dirichlet boundary conditions in space. Next, we define a family of periodic principal eigenvalues associated with the operator and use it to give a characterization of the generalized principal eigenvalue. Finally, we study the dependence of all these eigenvalues with respect to the coefficients.      

Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze-Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.     

Stable invariant manifolds for parabolic dynamics

We consider nonautonomous equations v^′=A(t)v in a Banach space that exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(t) for some function ρ(t). This corresponds to the existence of a “generalized” exponential dichotomy, which is known to be robust. When ρ(t)≠t this behavior can be described as a type of parabolic dynamics. We consider the general case of nonuniform exponential dichotomies, for which the Lyapunov stability is not uniform. We show that for any sufficiently small perturbation f of a “generalized” exponential dichotomy there is a stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v). We also consider the case of exponential contractions, which allow a simpler treatment, and we show that they persist under sufficiently small nonlinear perturbations.