An elliptical Gaussian wave formalism model of a charged-particle beam is proposed by analogy with an elliptical Gaussian light beam. In the paraxial approximation, the charged-particle beam can be described as a whol...An elliptical Gaussian wave formalism model of a charged-particle beam is proposed by analogy with an elliptical Gaussian light beam. In the paraxial approximation, the charged-particle beam can be described as a whole by a complex radius of curvature in the real space domains. Therefore, the propagation and transform of charged-particle beam passing through a first-order optical system is represented by the ABCD-like law. As an example of the application of this model, the relation between the beam waist and the minimum beam spot at a fixed target is discussed. The result, well matches that from conventional phase space model, and proves that the Gaussian wave formalism model is highly effective and reasonable.展开更多
In this paper, we show that any complete Riemannian manifold of dimension great than 2 must be compact if it has positive complex sectional curvature and δ-pinched 2-positive curvature operator, namely, the sum of th...In this paper, we show that any complete Riemannian manifold of dimension great than 2 must be compact if it has positive complex sectional curvature and δ-pinched 2-positive curvature operator, namely, the sum of the two smallest eigenvalues of curvature operator are bounded below by δ.scal 〉 O. If we relax the restriction of positivity of complex sectional curvature to non- negativity, we can also show that the manifold is compact under the additional condition of positive asymptotic volume ratio.展开更多
文摘An elliptical Gaussian wave formalism model of a charged-particle beam is proposed by analogy with an elliptical Gaussian light beam. In the paraxial approximation, the charged-particle beam can be described as a whole by a complex radius of curvature in the real space domains. Therefore, the propagation and transform of charged-particle beam passing through a first-order optical system is represented by the ABCD-like law. As an example of the application of this model, the relation between the beam waist and the minimum beam spot at a fixed target is discussed. The result, well matches that from conventional phase space model, and proves that the Gaussian wave formalism model is highly effective and reasonable.
文摘In this paper, we show that any complete Riemannian manifold of dimension great than 2 must be compact if it has positive complex sectional curvature and δ-pinched 2-positive curvature operator, namely, the sum of the two smallest eigenvalues of curvature operator are bounded below by δ.scal 〉 O. If we relax the restriction of positivity of complex sectional curvature to non- negativity, we can also show that the manifold is compact under the additional condition of positive asymptotic volume ratio.
