It is shown that the path topology of Hawking, King, and McCarthy can be extended to the causal completion of a globally hyperbolic Lorentzian manifold. The suggested topology T is defined only in terms of chronological structures and T is finer than the？

, in the : Applications and Reviews, 36(4):,？

We consider the topology of the BS causal completion, and show that it is not distorted in the causal direction. Using this result, we show that the causal completion with spacelike causal boundary in the sense of the BS construction satisfies the formal？

In this paper we are concerned with the existence and energy decay of solution to the initial boundary value problem for the coupled Klein–Gordon–Schrödinger equations with non­linear boundary damping and memory term.

Suppose that $P(D)$ is a linear differential operator with complex coefficients and $\{f_k\}_{k \in \mathbb{Z}}$ is a two-sided sequence of complex-valued functions on $\mathbb{R}$ such that $f_{k+1} \,{=}\, P(D)f_k$ with the following growth condition: there exist $a\,{\in}\,[0,1)$ and $N\,{\geq}\,0$ such that $|f_k(x)|\,{\leq}\,M_{|k|}\exp (N|x|^a)$, where the sequence $\{M_k\}_{k=0}^{\infty}$ satisfies that for any $\varepsilon\,{>}\,0$, the sequence $\{{M_k}/{(1+\varepsilon)^{k}}\}_{k=0}^{\infty}$ has a bounded subsequence. Then $f_0$ is an entire function of an exponential growth. This result is a generalization of those of Roe, Burkill and Howard.

We propose an electric resistivity inversion method that is similar to the reverse time migration technique applied to seismic data. For calculating model responses and inversion, we use the mixed finite-element method with the standard P₁ − P₀ pair for triangular decompositions, which makes it possible to compute both the electric potential and the electric field vector economically. In order to apply the adjoint state of the Poisson equation in the resistivity inverse problem, we introduce an apparent electric field defined as the dot product between the computed electric field vector and a weighting factor and then defining a virtual source to compute the partial derivative of the electric field vector. We exploit the adjoint state (the symmetry of Green’s function) of matrix equations derived from solving the Poisson equation by the mixed finite-element method, for the calculation of the steepest descent direction of our objective function. By computing the steepest descent direction by a dot product of backpropagated residual and virtual source, we can avoid the cumbersome and expensive process of computing the Jacobian matrix directly. We calibrate our algorithm on a synthetic of a buried conductive block and obtain an image that is compatible with the limits of the resistivity method.

We present nonoverlapping domain decomposition methods for the approximation of both electromagnetic fields in a three-dimensional bounded domain satisfying absorbing boundary conditions. {\it A Seidel-type domain decomposition iterative method} is introduced based on a hybridization of a {\it nonconforming mixed finite element method.} Convergence results for the numerical procedure are proved by introducing a suitable pseudo-energy. The spectral radius of the iterative procedure is estimated and a method for choosing an optimal matching parameter is given. A red-black Seidel-type method which is readily parallelizable is also introduced and analyzed. Numerical experiments confirm that the presented algorithms are faster than the conventional {\it Jacobi-type ones}.