Patrick Speissegger
To illustrate how we do this, let us do this for the input search algorithm, both for the Classical Turing machine (CTM) and the Probabilistic Turing Machine (PTM). In both cases, it makes sense to express the number of steps needed to find the input as functions $t_{\textrm{CTM}}(d)$ and $t_{\textrm{PTM}}(d)$ of the distance $d$ of…
In this appendix to the Fields 2022 module, I briefly discuss a couple of loose ends left in this post. First, let $f = (f_0, \dots, f_k) \in \U^{k+1}$ be such that $f_0 > \cdots > f_k$. We saw that if $f_0 \succ \cdots \succ f_k$, then our construction yields a qaa field $(K_f, e^x…
(Joint work with Zeinab Galal and Tobias Kaiser) I describe here the generalization of this construction to arbitrary convergent transmonomials, based on the Continuation Theorem. Definition A germ $f \in \I$ is simple if $\level(f) = \eh(f)$. For example, every convergent transmonomial is simple, as is every purely infinite germ. Recall that $$\la := \bigcup_{n…
(Joint work with Tobias Kaiser) I introduce real domains in $\LL$ and angular level, and I use these notions to describe holomorphic continuations on $\LL$ of one-variable functions definable in $\Ranexp$. In this post “definable” means “definable in $\Ranexp$”. DefinitionA set $U \subseteq \LL$ is a real domain if there exist $a \gt 0$ and…
(Joint work with Tobias Kaiser; this is a repost) We are interested in holomorphic continuations of one-variable functions definable in $\Ranexp$. Since $\exp$ and $\log$ are two crucial functions definable in $\Ranexp$, the natural domain on which to consider holomorphic continuations of all definable functions is the Riemann surface of the logarithm $$\LL:= (0,\infty) \times…