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2018-05-29 Nonlinear Seminar

Xiao-Xiong Gan (Morgan State University, Baltimore MD, USA)
From Formal Power Series to Formal Analysis

For any $$ l \in \mathbb{N} $$, a formal power series on a ring $$ S $$ is defined to be a mapping from $$ \mathbb{N}^l $$ to $$ S $$. A formal power series $$ f $$ in $$ x $$ from $$ \mathbb{N} $$ to $$ S $$ is usually denoted as a sequence $$ (a_0, a_1, a_2, \dots) $$ or as a power series $$ f(z) = a_0 + a_1 z + \cdots + a_n z^n + \cdots $$, where $$ a_j \in S $$ for every $$ j \in \mathbb{N}\cup\{0\} $$. The set of all formal power series on $$ S $$ is denoted by $$ \mathbb{X}(S) $$.

If considering a formal power series as a sequence, what is the difference between $$ \mathbb{X} $$ and $$ l^p $$?

If considering a formal power series as a power series, what is the difference between formal power series and the traditional power series? What is the relationship between formal power series and traditional power series?

What is formal analysis?

This talk tries to answer those questions and brings discussion of all kinds of questions about formal analysis, a relatively new mathematical subject.