Barbara Zwi knagl1 and Robert S haba k2 Abstra t: The univariate Taylor formula without remainder allows to reprodu e a fun tion ompletely from ertain derivative values. Thus one an look for Hilbert spa es in whi h the Taylor formula a ts as a reprodu tion formula. It turns out that there are many Hilbert spa es whi h allow this, and they should be alled Taylor spa es. They have ertain reprodu ...

In this talk, we consider a class of nonautonomous difference equations (recursions) possessing an attractive invariant manifold in the extended state space. It is our aim to study the behavior of this manifold, as well as of its corresponding asymptotic phase under perturbation. Our framework is sufficiently general to include discrete counterparts of (center-)unstable and inertial manifolds. ...

In the setting of exponential investors and uncertainty governed by Brownian motions we first prove the existence of an incomplete equilibrium for a general class of models. We then introduce a tractable class of exponential-quadratic models and prove that the corresponding incomplete equilibrium is characterized by a coupled set of Riccati equations. Finally, we prove that these exponential-qu...

The IBM Power4 processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theorem, and then analyzing the Chebyshev series using Taylor series. Taylor’s theorem is proved by way of non-stand...

Methods of local approximations based on truncated Taylor series are well understood; several methods to compute bounds for the approximation errors are available (Christensen and Christensen, 2006), (Powell, 1981), (Shahriari, 2006). The use of Taylor approximations is convenient, among others, because methods to determine the approximation accuracy are well established. Taylor approximations ...

The Taylor model [8] is one of the inclusion functions available to compute the range enclosures. It has the property of (m+ 1) convergence order, where, m is the order of the Taylor model used. It computes a high order polynomial approximation to a multivariate Taylor expansion, with a remainder term that rigorously bound the approximation error. The sharper bounds on the enclosures computed u...

A particle method is presented for computing vortex sheet motion in three-dimensional flow. The particles representing the sheet are advected by a regularized Biot– Savart integral in which the exact singular kernel is replaced by the Rosenhead– Moore kernel. New particles are inserted to maintain resolution as the sheet rolls up. The particle velocities are evaluated by an adaptive treecode al...

This paper proposes an analytical approximation to price exotic options within a stochastic volatility framework. Assuming a general mean reverting process for the underlying asset and a square-root process for the volatility, we derive an approximation for option prices using a Taylor expansion around two average defined volatilities. The moments of the average volatilities are computed analyt...

As two-dimensional coupled system of nonlinear partial differential equations does not give enough smooth solutions, when approximated by linear, quadratic and cubic polynomials and gives poor convergence or no convergence. In such cases, approximation by zero degree polynomials like Haar wavelets (continuous functions with finite jumps) are most suitable and reliable. Therefore, modified numer...