Summarizing and completing results by many mathematicians during the past four decades, Fishman, Simmons, and Urbanski provide a complete theory of Diophantine

approximation in the limit set of a group acting on a Gromov hyperbolic metric space.

Keywords: NP completeness,

approximation algorithm, k-median problem

On the basis of existence of parameter manifold, the numeration

approximation systems of iterative levels have been given by backward-forward systems.

The aim of this work is to present an analytical

approximation study of periodic solutions for systems of second-order nonlinear differential equations.

Now we say that the discrete time

approximation [x.sub.h] with the step-size h converges strongly of order y at time T = Nh to the solution X(t) if

They discussed Korovkin-type

approximation properties and rate of convergence of operators (2).

Key words: cosine operator function, Blackman-type

approximation processes, Rogosinski-type

approximation processes, modulus of continuity, Fourier series of symmetric functions with respect to [pi].

The first-order saddlepoint

approximation (FOSA) presented in [3], is based on the

approximation of the limit state function at the most likelihood point in the original (not standard normal) space.

An efficient initial

approximation was proposed in [3], although it concentrated on division and square root, it can lead to a solution of inverse square root as well.

A second-order

approximation is obtained by assuming that the change in molar volume with pressure can be described by a constant isothermal compressibility, [[kappa].sub.T] = -1/[v.sub.w][([partial derivative][v.sub.w]/[partial derivative]p).sup.T], which is taken as its value at saturation, [[kappa].sup.sat.sub.T].

The function [[phi].sub.a](x, [epsilon]) is named as asymptotic

approximation of the function [phi](x, [epsilon]).

As an application, for the particular case of CEV model, we obtain an

approximation of the at-the-money (ATM) implied volatility curve as a function of time and an

approximation of the implied volatility smile as a function of the log-moneyness, close to the expiry date.