The angular correlation of the two gamma quanta emitted when a thermalised positron annihilates with metallic conduction electrons is investigated by applying the newly developed theory in which an electron gas is considered as a system of interacting collective excitations. The conduction electrons are treated in a jellium model. The method leads in a natural way to the appearance of high-momentum components (i.e. pair momentum P > pF) in the annihilation radiation. The amount of these components is significant ( 2 lo",) in a dilute electron gas (such as in alkali metals), but fairly irrelevant for higher densities. The dependence on momentum of the enhancement factor for a dense system (wzith r, = 2 ) agrees well both with the earlier theories given b) Kahana and others, and also with recent accurate experimental observations. As r, increases into the alkali-metal region. the enhancement factor for P c. pF becomes relatively more and more constant, in contrast with the trend in the Kahana theory. In this density regime the experimental results vary widely. although most of them disagree with the present prediction. The additional effects due to core annihilation and lattice periodicity are studied and reviewed.
The wave pattern created by a pressure point moving with constant velocity over the free surface of an inviscid MHD plasma bordered by a magnetic field is investigated. If the depth of the plasma is taken to be Mnite, the results are the same as found in the case of an ordinary fluid provided that the velocity of the pressure point exceeds the Alfvbvelocity. In this case a new expression is given for the surface elevation in front of the pressure point. Moreover some new results are found in the cases of a fluid and a plasma of iinite depth.
ABSTRACT. We prove that to any invariant subset of the dynamical system generated by a one-dimensional quasilinear parabolic equation there corresponds an invariant family of stable manifolds of finite codimension.K~.v WORDS: infinite-dimensional dynamical systems, asymptotic behavior of trajectories, invariant families of stable manifolds.
w IntroductionThe asymptotic behavior of trajectories is one of the main topics in the theory of evolution differential equations. In particular, it can be studied by constructing invariant families of manifolds. In the present paper, we prove the existence of a family of stable manifolds associated with an invariant subset of the dynamical system generated by the one-dimensional parabolic equation The existence of invariant manifolds for infinite-dimensional dynamical systems generated by equations of the form (1) was studied in [2][3][4] under the assumption that f does not depend on uz. This restriction is removed in the present paper. To make the exposition shorter, we consider only autonomous equations; this simplification is not essential, and all results can readily be generalized to the nonautonomous case. To be definite, let us consider the homogeneous Dirichlet boundary conditions. w The dynamical system Consider the Hilbert space W of square integrable functions on [0,1]. The space W is equipped with the standard inner product (., 9 ), which generates the corresponding norm [ 9 ] [1]. The operator L is densely defined and self-adjoint in W. Furthermore, there exists an orthonormal basis {vk(z)}, k E N, in W such that Lvk = A~v~, A~ < A~+l, and A~ --* oo.For each integer k we define a Hilbert space W k as the completion of the domain of L k/2 with respect to the norm I lk = ILk/ l, generated by the inner product (u, v)k = (Lk/2u, L~/2v). We set W ~ and W~176 N Wk k>_l by definition. By the Rellich theorem, the natural embedding W k --* W k-1 is compact.