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Andrew Vince

Publications and source records attributed to Andrew Vince.

29 records · Page 2Linked to original sources
arXiv2013

The Entropy of an Overlapping Dynamical System

The term "overlapping" refers to a certain fairly simple type of piecewise continuous function from the unit interval to itself and also to a fairly simple type of iterated function system (IFS) on the unit interval. A correspondence between these two classes of objects is used (1) to find a necessary and sufficient condition for a fractal transformation from the attractor of one overlapping IFS to the attractor of another overlapping IFS to be a homeomorphism and (2) to find a formula for the topological entropy of the dynamical system associated with an overlapping function.

math.DS↗
arXiv2012

Fractal Continuation

A fractal function is a function whose graph is the attractor of an iterated function system. This paper generalizes analytic continuation of an analytic function to continuation of a fractal function.

math.DS↗
arXiv2012

The Conley Attractor of an Iterated Function System

We investigate the topological and metric properties of attractors of an iterated function system (IFS) whose functions may not be contractive. We focus, in particular, on invertible IFSs of finitely many maps on a compact metric space. We rely on ideas Kieninger and McGehee and Wiandt, restricted to what is, in many ways, a simpler setting, but focused on a special type of attractor, namely point-fibred minimal (locally) invariant sets. This allows us to give short proofs of some of the key ideas.

math.DS↗
arXiv2012

The Number of Ways to Assemble a Graph

Motivated by the question of how macromolecules assemble, the notion of an {\it assembly tree} of a graph is introduced. Given a graph $G$, the paper is concerned with enumerating the number of assembly trees of $G$, a problem that applies to the macromolecular assembly problem. Explicit formulas or generating functions are provided for the number of assembly trees of several families of graphs, in particular for what we call $(H,ϕ)$-graphs. In some natural special cases, we apply powerful recent results of Zeilberger and Apagodu on multivariate generating functions, and results of Wimp and Zeilberger, to deduce recurrence relations and very precise asymptotic formulas for the number of assembly trees of the complete bipartite graphs $K_{n,n}$ and the complete tripartite graphs $K_{n,n,n}$. Future directions for reseach, as well as open questions, are suggested.

math.CO↗
arXiv2011

Fractal Homeomorphism for Bi-affine Iterated Function Systems

The paper concerns fractal homeomorphism between the attractors of two bi-affine iterated function systems. After a general discussion of bi-affine functions, conditions are provided under which a bi-affine iterated function system is contractive, thus guaranteeing an attractor. After a general discussion of fractal homeomorphism, fractal homeomorphisms are constructed for a specific type of bi-affine iterated function system.

math.DS↗
arXiv2011

Real Projective Iterated Function Systems

This paper contains four main results associated with an attractor of a projective iterated function system (IFS). The first theorem characterizes when a projective IFS has an attractor which avoids a hyperplane. The second theorem establishes that a projective IFS has at most one attractor. In the third theorem the classical duality between points and hyperplanes in projective space leads to connections between attractors that avoid hyperplanes and repellers that avoid points as well as hyperplane attractors that avoid points and repellers that avoid hyperplanes. Finally, an index is defined for attractors which avoid a hyperplane. This index is shown to be a nontrivial projective invariant.

math.DS↗
arXiv2010

The Chaos Game on a General Iterated Function System

The main theorem of this paper establishes conditions under which the "chaos game" algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non contractive iterated function systems, and under weaker conditions on the random orbit of the chaos game than obtained previously.

math.MG↗
arXiv2010

The Eigenvalue Problem for Linear and Affine Iterated Function Systems

The eigenvalue problem for a linear function L centers on solving the eigen-equation Lx = rx. This paper generalizes the eigenvalue problem from a single linear function to an iterated function system F consisting of possibly an infinite number of linear or affine functions. The eigen-equation becomes F(X) = rX, where r>0 is real, X is a compact set, and F(X)is the union of f(X), for f in F. The main result is that an irreducible, linear iterated function system F has a unique eigenvalue r equal to the joint spectral radius of the functions in F and a corresponding eigenset S that is centrally symmetric, star-shaped, and full dimensional. Results of Barabanov and of Dranishnikov-Konyagin-Protasov on the joint spectral radius follow as corollaries.

math.MG↗
arXiv2009

A Characterization of Hyperbolic Affine Iterated Function Systems

The two main theorems of this paper provide a characterization of hyperbolic affine iterated function systems defined on Rm. Atsushi Kameyama (Distances on Topological Self-Similar Sets, Proceedings of Symposia in Pure Mathematics, Volume 72.1, 2004) asked the following fundamental question: given a topological self-similar set, does there exist an associated system of contraction mappings? Our theorems imply an affirmative answer to Kameyama's question for self-similar sets derived from affine transformations on Rm.

math.GT↗
arXiv2009

Tree Orbits under Permutation Group Action: Algorithm, Enumeration and Application to Viral Assembly

This paper uses combinatorics and group theory to answer questions about the assembly of icosahedral viral shells. Although the geometric structure of the capsid (shell) is fairly well understood in terms of its constituent subunits, the assembly process is not. For the purpose of this paper, the capsid is modeled by a polyhedron whose facets represent the monomers. The assembly process is modeled by a rooted tree, the leaves representing the facets of the polyhedron, the root representing the assembled polyhedron, and the internal vertices representing intermediate stages of assembly (subsets of facets). Besides its virological motivation, the enumeration of orbits of trees under the action of a finite group is of independent mathematical interest. If $G$ is a finite group acting on a finite set $X$, then there is a natural induced action of $G$ on the set $\mathcal{T}_X$ of trees whose leaves are bijectively labeled by the elements of $X$. If $G$ acts simply on $X$, then $|X| := |X_n| = n \cdot |G|$, where $n$ is the number of $G$-orbits in $X$. The basic combinatorial results in this paper are (1) a formula for the number of orbits of each size in the action of $G$ on $\mathcal{T}_{X_n}$, for every $n$, and (2) a simple algorithm to find the stabilizer of a tree $τ\in \mathcal{T}_X$ in $G$ that runs in linear time and does not need memory in addition to its input tree.

math.CO↗