ON THE VERTEX PARTITION EQUATION OF ROOTED LOOPLESS PLANAR MAPS

This paper provides a functional equation astisfied by the generating function for enumerating rooted loopless planar maps with vertex partition. A kind of applications in enumerating, by providing explicit formulae, a type of rooted loopless planar maps with the maximum valency of vertices given are described. Meanwhile, the functional equation for enumerating rooted loopless planar maps (connected) with the edge number and the valency of root-vertex as the parameters is also derived directly.

High-Order Iterative Methods Repeating Roots a Constructive Recapitulation

This paper considers practical, high-order methods for the iterative location of the roots of nonlinear equations, one at a time. Special attention is being paid to algorithms also applicable to multiple roots of initially known and unknown multiplicity. Efficient methods are presented in this note for the evaluation of the multiplicity index of the root being sought. Also reviewed here are super-linear and super-cubic methods that converge contrarily or alternatingly, enabling us, not only to approach the root briskly and confidently but also to actually bound and bracket it as we progress.

The Offset-Domain Prestack Depth Migration with Optimal Separable Approximation

The offset-domain prestack depth migration with optimal separable approximation, based on the double square root equation, is used to image complex media with large and rapid velocity variations. The method downward continues the source and the receiver wavefields simultaneously. The mixed domain algorithm with forward Fourier and inverse Fourier transform is used to construct the double square root equation wavefield extrapolation operator. This operator separates variables in the wave number domain and variables in the space domain. The phase operation is implemented in the wave number domain, whereas the time delay for lateral velocity variation is corrected in the space domain. The migration algorithm is efficient since the seismic data are not computed shot by shot. The data set test of the Marmousi model indicates that the offset-domain migration provides a satisfied seismic migration section on which complex geologic structures are imaged in media with large and rapid lateral velocity variations.

On solving equations of algebraic sum of equal powers

It is well known that a system of equations of sum of equal powers can be converted to an algebraic equation of higher degree via Newton's identities. This is the Viete-Newton theorem. This work reports the generalizations of the Viete-Newton theorem to a system of equations of algebraic sum of equal powers. By exploiting some facts from algebra and combinatorics,it is shown that a system of equations of algebraic sum of equal powers can be converted in a closed form to two algebraic equations, whose degree sum equals the number of unknowns of the system of equations of algebraic sum of equal powers.

ON STABILITY OF SYMPLECTIC ALGORITHMS

The stability of symplectic algorithms is discussed in this paper. There are following conclusions. 1. Symplectic Runge-Kutta methods and symplectic one-step methods with high order derivative are unconditionally critically stable for Hamiltonian systems. Only some of them are A-stable for non-Hamiltonian systems. The criterion of judging A-stability is given. 2. The hopscotch schemes are conditionally critically stable for Hamiltonian systems. Their stability regions are only a segment on the imaginary axis for non-Hamiltonian systems. 3. All linear symplectic multistep methods are conditionally critically stable except the trapezoidal formula which is unconditionally critically stable for Hamiltonian systems. Only the trapezoidal formula is A-stable, and others only have segments on the imaginary axis as their stability regions for non-Hamiltonian systems.