Related papers: Analytic and Topological Nets

Let $f:S^2\to S^2$ be an orientation-preserving branched covering map of degree $d\geq 2$, and let $\Sigma$ be an oriented Jordan curve passing through the critical values of $f$. Then $\Gamma:=f^{-1}(\Sigma)$ is an oriented graph on the…

Let C be real-analytic simple closed curve in the complex plane which is symmetric with respect to the real axis. Let r>0 be such that C+ir misses C-ir. We prove that if a continuous function f extends holomorphically from C+it for each t…

Under some mild assumptions, an orientation-preserving branched covering map of marked $2$-spheres induces a pullback map between the corresponding Teichm\"uller spaces. By analyzing the associated pushforward operator acting on integrable…

We characterize the space of restrictions of real rational functions to certain algebraic Jordan curves in the plane via the Dirichlet-to-Neumann map associated to the domain in the complex plane bounded by the curve and its Bergman kernel.…

We investigate automorphism groups of planar graphs. The main result is a complete recursive description of all abstract groups that can be realized as automorphism groups of planar graphs. The characterization is formulated in terms of…

Let $f$ be a meromorphic function with simply connected domain $G\subset\mathbb{C}$, and let $\Gamma\subset\mathbb{C}$ be a smooth Jordan curve. We call a component of $f^{-1}(\Gamma)$ in $G$ a $\Gamma$-$pseudo$-$lemniscate$ of $f$. In this…

A special class of Jordan algebras over a field $F$ of characteristic zero is considered. Such an algebra consists of an $r$-dimensional subspace of the vector space of all square matrices of a fixed order $n$ over $F$. It contains the…

Let f: P^1 \to P^1 be a rational map with finite postcritical set P_f. Thurston showed that f induces a holomorphic map \sigma_f of the Teichmueller space T modelled on P_f to itself fixing the basepoint corresponding to the identity map…

We consider rational maps $f$ on the Riemann sphere $\widehat {\mathbb{C}}$ with an $f$-invariant set $P\subset \widehat {\mathbb{C}}$ of four marked points containing the postcritical set of $f$. We show that the dynamics of the…

We define pullback and separated presentations of modules over pullback rings, and, if the ring is a pullback of epimorphisms over a semisimple ring, an algorithm reducing such a presentation of a module to an $R$-diagram. The latter is the…

In Functional Analysis, certain conclusions apply to sequences, but they cannot be carried over when we consider nets. In fact, some nets, including sequences, can behave unexpectedly. In this paper we are interested in exploring the…

Algebraic domains are regions in the plane surrounded by mutually disjoint non-singular real algebraic curves. Poincar'e-Reeb Graphs of them are graphs they naturally collapse: such graphs are formally formulated by Sorea, for example,…

We develop a categorical framework for studying graphs of groups and their morphisms, with emphasis on pullbacks. More precisely, building on classical work by Serre and Bass, we give an explicit construction of the so-called…

Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Finite collections of disjoint Jordan domains can be approximated by the basins of attraction of rational…

Our main result states that, under an exponential map whose Julia set is the whole complex plane, on each piecewise smooth Jordan curve there is a point whose orbit is dense. This has consequences for the boundaries of nice sets, used in…

The existence of a finite global attractor for polynomial curve system has been known since the work of Belk et al. [4]. However, except in the hyperbolic case, the rate at which the pullback of a curve under a polynomial converges to the…

Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph $\Lambda$ is a natural generalization of a higher rank graph. A pullback of $\Lambda$ is constructed by pulling it back over a given monoid morphism to $P$, while a…

A rational lemniscate is a level set of $|r|$ where $r: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}$ is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This…

Consider a rational map from a projective space to a product of projective spaces, induced by a collection of linear projections. Motivated by the the theory of limit linear series and Abel-Jacobi maps, we study the basic properties of the…

We use the framework of "props" to study electrical circuits, signal-flow diagrams, and bond graphs. A prop is a strict symmetric monoidal category where the objects are natural numbers, with the tensor product of objects given by addition.…