Taylor series

Summary

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a func

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In this thesis, we give a modern treatment of Dwyer's tame homotopy theory using the language of

\infty

-categories.We introduce the notion of tame spectra and show it has a concrete algebraic description.We then carry out a study of

\infty

-operads and define tame spectral Lie algebras and tame spectral Hopf algebras. Finally, we prove that the homotopy theory of tame spectral Hopf algebras is equivalent to that of tame spaces. To recover Dwyer's Lie algebra model for tame spaces, we use Koszul duality to construct a universal enveloping algebra functor, and show it is an equivalence from the

\infty

-category of tame spectral Lie algebras to the

\infty

-category of tame spectral Hopf algebras.

EPFL2022

Perturbation analysis for the Darcy problem with log-normal permeability

Francesca Bonizzoni, Fabio Nobile

We study the single-phase flow in a saturated, bounded heterogeneous porous medium. We model the permeability as a log-normal random field. We perform a perturbation analysis, expanding the solution in Taylor series. The series is directly computable in the case of a random field parametrized by a finite number of random variables. On the other hand, in the case of an infinite dimensional random field, suitable equations satisfied by the derivatives of the stochastic solution can be derived. We give a theoretical upper bound for the norm of the residual of the Taylor expansion which predicts the divergence of the series as the polynomial degree goes to infinity. We provide a formula to compute the optimal degree for the Taylor polynomial and the maximum achievable accuracy of the perturbation approach. Our theoretical findings are confirmed by numerical experiments in the simple case where the permeability field is described using one random variable

2014

MATHICSE Technical Report : Perturbation analysis for the Darcy problem with log-normal permeability