An electron in the field of two classical plane waves propagating in slightly different directions

Solutions of the Dirac and Klein-Gordon equations have been obtained for an electron moving in the field of two plane waves propagating in slightly different directions. The solutions were obtained as an expansion in terms of the small angle between the two waves. Terms linear in this angle were considered.     

The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C ^1·ω (R ⁿ )to an arbitrary closed subset of R ⁿ .The similar result is obtained for some other spaces of multivariate smooth functions. We also show that unlike the one-dimensional case treated by Whitney, for some trace spaces of multivariate smooth functions a linear bounded extension operator does not exist. The proofs are based on a relation between the problem under consideration and a similar problem for Lipschitz spaces defined on hyperbolic Riemannian manifolds.     

The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

We study a variant of the Whitney extension problem (1934) for the space . We identify with a space of Lipschitz mappings from into the space of polynomial fields on equipped with a certain metric. This identification allows us to reformulate the Whitney problem for as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of . We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for due to C. Fefferman.

     

Necessary Optimality Conditions in Discrete Nonsmooth Optimal Control

In this paper, we provide a simple proof of the maximum principle for a nonsmooth discrete-time optimal control problem. The methodology is general and encompasses all generalized derivatives for which the Lagrange multiplier rule and the chain rule hold. This includes, but is not limited to, limiting (Mordukhovich) and Michel–Penot subdifferentials.     

The suppression of selected acoustic frequencies in MRI

A study is made of certain dominant frequencies in the acoustic noise spectrum of the magnetic resonance imaging system. Motivated by both spring and string ideas, we investigate whether the contributions to the sound from certain frequencies can be canceled by the appropriate gradient pulse sequence design. From both simulations and experiments, vibrations resulting from an impulsive force associated with a ramping up of a gradient pulse are shown to be cancelled immediately upon the application of another impulsive force coming from the subsequent appropriately timed ramping down of that pulse. A general approach to suppression of multiple-frequency contributions involving a series of gradient pulses with variable timings is given for the cancellations between pairs of impulsive forces. Various examples are confirmed through string simulations, MRI experiments, and linear response theory. This also provides a foundation to explain some results in previous papers on this subject. The method suggests that a variety of pulse profiles and timing combinations can be used to attenuate important contributions to the acoustic spectrum.     

Phase states and ferroelectric properties of Pb(Zr, Sn, Ti)O3 ceramic

The ferroelectric and dielectric properties of Pb_0.99[(Zr_1−x Sn_x)_1−y Ti_y]_0.98Nb_0.02O₃ ceramic of composition (I) x=0.5, y=0.04–0.12 and (II) x=0.4, y=0.06–0.09 were studied, and the T-x and E-T diagrams were constructed. When zinc is substituted for tin, the range of coexistence of the ferroelectric and antiferroelectric phases extends. The range of the nonpolar phase shrinks when the titanium content is increased.