Patrick Speissegger
This post generalizes the construction of quasianalytic Ilyashenko algebras based on log monomials to certain other definable monomials. This construction is joint work with my student Zeinab Galal. Recall that $\H = \Hanexp$ is the Hardy field of germs at $+\infty$ of univariate functions definable in $\Ranexp$, and that $\I$ is the set of all…
Can the 0 gate be built as a circuit using only reversible gates? in other words, is there a classical circuit $C$ of width $n$ (unkown), built entirely from reversible gates, such that its first wire produces output 0 no matter what the input is? The answer is no! To see why, let’s look at…
This post describes the construction of quasianalytic algebras of functions with simple logarithmic transseries as asymptotic expansions. The construction is based on Ilyashenko’s class of almost regular functions, as introduced in his book on Dulac’s problem. This class forms a group under composition, but it is not closed under addition or multiplication; to obtain a…
To wrap up the notes on o-minimal structures, let’s consider two questions that were still open at the end of the last millennium: Questions Does every o-minimal expansion of the real field admit analytic cell decomposition? Is there a unique maximal o-minimal expansion of the real field? The answer to both questions is “no”, and…
Let $\Delta_n$ be a collection of subsets of $\RR^n$, for $n \in \NN$, and set $\Delta = (\Delta_n)_{n \in \NN}$. As usual, we refer to the elements of the various $\Delta_n$ as $\Delta$-sets, and we call $\Delta$-sets that are manifolds $\Delta$-manifolds. We assume the following axioms for $\Delta$-sets: $(\Delta 1)$ every semialgebraic set is a…