arXiv subjects
Publications and source records attributed to Roman Kogan.
We study the images of the Markov measures under transformations generated by the Mealy automata. We find conditions under which the image measure is absolutely continuous or singular relative to the Markov measure. Also, we determine statistical properties of the image of a generic sequence.
We introduce the notion of the Automatic Logarithm $\mathcal L_{\mathcal A, \mathcal B}$ with the purpose of studying the expanding properties of Schreier graphs of action of the group generated by two finite initial Mealy automata $\mathcal A$ and $\mathcal B$ on the levels of a regular $d$-ary rooted tree $\mathcal T$, where $\mathcal A$ is level-transitive and of bounded activity. $\mathcal L_{\mathcal A, \mathcal B}$ computes the lengths of chords in this family of graphs. Formally, $\mathcal L$ is a map $\partial \mathcal T \rightarrow \mathbb{Z}_d$ from the boundary of the tree to the integer $p$-adics whose values are determined by a Moore machine. The distribution of its outputs yields a probabilistic measure $\mu$ on $\partial \mathcal T$, which in some cases can be computed by a Mealy-type machine (we then say that $\mu$ is finite-state). We provide a criterion to determine whether $\mu$ is finite-state. A number of examples illustrating the different cases with $\mathcal A$ being the adding machine is provided.
Given any complex Laurent polynomial $f$, $\mathrm{Amoeba}(f)$ is the image of its complex zero set under the coordinate-wise log absolute value map. We give an efficiently constructible polyhedral approximation, $\mathrm{ArchtTrop}(f)$, of $\mathrm{Amoeba}(f)$, and derive explicit upper and lower bounds, solely as a function of the number of monomial terms of $f$, for the Hausdorff distance between these two sets. We also show that deciding whether a given point lies in $\mathrm{ArchTrop}(f)$ is doable in polynomial-time, for any fixed dimension, unlike the corresponding problem for $\mathrm{Amoeba}(f)$, which is $\mathbf{NP}$-hard already in one variable. $\mathrm{ArchTrop}(f)$ can thus serve as a canonical low order approximation to start any higher order iterative polynomial system solving algorithm, such as homotopy continuation. $\mathrm{ArchTrop}(f)$ also provides an Archimedean analogue of Kapranov's Non-Archimedean Amoeba Theorem and a higher-dimensional extension of earlier estimates of Mikhalkin and Ostrowski.