Andrew Vince

University of Florida | UF · Department of Mathematics

Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. Moreover, contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Th...

Let $G$ denote the infinite grid graph with vertex set $\{(a,b)\ : \, a,b \in \mathbb{Z}\}$ and edge set $\big \{ \{u,v\} : |u-v|=1 \;\text{or}\; |u-v| = \sqrt{2} \big \}.$ A question in landscape ecology, restated in graph theoretic terms, asks the following. What is the maximum number of edges in an induced subgraph of $G$ of order $n$? It was co...

It is proved that homeomorphic images of certain two-dimensional aperiodic tilings, such as Ammann A2 tilings, are recognizable, in both mathematical and practical senses. One implication of the results is that it is possible to search for distorted aperiodic structures in nature, where they may be hiding in plain sight.

Because connectivity is such a basic concept in graph theory, extremal problems concerning the average order of the connected induced subgraphs of a graph have been of notable interest. A particularly resistant open problem is whether or not, for a connected graph of order , all of whose vertices have degree at least 3, this average is at least . I...

Iterated function systems (IFSs) and their attractors have been central in fractal geometry. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Two natural questions concerning contractivity arise. First, whether an IFS needs to be contractive to admit an attractor? Second, what occurs to the attrac...

The topic is the average order of a connected induced subgraph of a graph. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1983, Jamison proved that the average order of a subtree, over all trees of order n, is minimized by the path Pn. In 2018, Kroeker, Mol, and Oellermann conjectured that Pn minimizes the aver...

It is proved that homeomorphic images of certain two-dimensional aperiodic tilings, such as Ammann-A2 tilings, are recognizable, in both mathematical and practical senses. One implication of the results is that it is possible to search for distorted aperiodic structures in nature, where they may be hiding in plain sight.

The topic is the average order A(G) of a connected induced subgraph of a graph G. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1983 Jamison proved that the average order, over all trees of order n, is minimized by the path Pn, the average being A(Pn)=(n+2)/3. In 2018, Kroeker, Mol, and Oellermann conjectured...

A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system (GIFS), based on a combinatorial structure we call a pre-tree, is introduced. For each GIFS, a family of tilings is constructed indexed by a parameter. For what we call a commensurate GIFS, our method is used to define what we re...

The topic is the average order $A(G)$ of a connected induced subgraph of a graph $G$. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1984, Jamison proved that the average order, over all trees of order $n$, is minimized by the path $P_n$, the average being $A(P_n)=(n+2)/3$. In 2018, Kroeker, Mol, and Oellermann...

The topic is the average order of a connected induced subgraph of a graph. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1984, Jamison proved that the average order, over all trees of order $n$, is minimized by the path $P_n$. In 2018, Kroeker, Mol, and Oellermann conjectured that $P_n$ minimizes the average o...

Although connectivity is a basic concept in graph theory, the enumeration of connected subgraphs of a graph has only recently received attention. The topic of this paper is the average order of a connected induced subgraph of a graph. This generalizes, to graphs in general, the average order of a subtree of a tree. For various infinite families of...

This paper examines thresholds for certain properties of the attractor of a general one-parameter affine family of iterated functions systems. As the parameter increases, the iterated function system becomes less contractive, and the attractor evolves. Thresholds are studied for the following properties: the existence of an attractor, the connectiv...

This paper provides an approach to the study of self-similar tilings and substitution tilings, in the setting of graph-directed iterated function systems, where the tiles may be fractals and the tiled set maybe a complicated unbounded subset of $\mathbb{R}^{M}$.

This paper examines thresholds for certain properties of the attractor of a general one-parameter affine family of iterated functions systems. As the parameter increases, the iterated function system becomes less contractive, and the attractor evolves. Thresholds are studied for the following properties: the existence of an attractor, the connectiv...

A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system, based on a combinatorial structure we call a pre-tree, is introduced. In the special case that we refer to as balanced, the resulting tilings have a finite set of prototiles, are quasiperiodic but not periodic, and are self-simi...

The ground set for all matroids in this paper is the set of all edges of a complete graph. The notion of a {\it maximum matroid for a graph} $G$ is introduced, and the existence and uniqueness of the maximum matroid for any graph $G$ is proved. The maximum matroid for $K_3$ is shown to be the cycle (or graphic) matroid. This result is pursued in tw...

The concepts of a splicing machine and of an aparalled digraph are introduced. A splicing machine is basically a means to uniquely obtain all circular sequences on a finite alphabet by splicing together circular sequences from a small finite set of “generators”. The existence and uniqueness of the central object related to an aparallel digraph, the...

The attractor is a central object of an iterated function system (IFS), and fractal transformations are the natural maps from the attractor of one IFS to the attractor of another. This paper presents a global point of view, showing how to extend the domain of a fractal transformation from an attractor with non-empty interior to the ambient space. I...

A novel method for the construction of self-similar polygonal tilings, based on la- beled rooted trees, will be discussed.

The theory of fractal tilings of fractal blow-ups is extended to graph-directed iterated function systems, resulting in generalizations and extensions of some of the theory of Anderson and Putnam and of Bellisard et al. regarding self-similar tilings.

New tilings of certain subsets of $\mathbb{R}^{M}$ are studied, tilings associated with fractal blow-ups of certain similitude iterated function systems (IFS). For each such IFS with attractor satisfying the open set condition, our construction produces a usually infinite family of tilings that satisfy the following properties: (1) the prototile se...

This paper concerns two related enumeration problems on vertex labeled graphs. Given such a graph G, we investigate the number C(G) of connected subsets of the vertex set and the number P(G) of connected partitions of the vertex set. By connected we mean that the induced subgraphs are connected. The numbers C(G) and P(G) can be regarded as the (con...

The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting, and to provide a novel method for the construction of such polygonal tilings.

The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting, and to provide a novel method for the construction of such polygonal tilings.

Our results and examples show how transformations between self-similar sets may be continuous almost everywhere with respect to measures on the sets and may be used to carry well known notions from analysis and functional analysis, for example flows and spectral analysis, from familiar settings to new ones. The focus of this paper is on a number of...

Many remarkably robust, rapid and spontaneous self-assembly phenomena in nature can be modeled geometrically starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and bunches could be identical as well, the underlying sym...

Many remarkably robust, rapid and spontaneous self-assembly phenomena in nature can be modeled geometrically starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and bunches could be identical as well, the underlying sym...

For given $p\in\lbrack1,\infty]$ and $g\in L^{p}\mathbb{(R)}$, we establish the existence and uniqueness of solutions $f\in L^{p}(\mathbb{R)}$, to the equation \begin{equation*} f(x)-af(bx)=g(x), \end{equation*} where $a\in\mathbb{R}$, $b\in\mathbb{R},$ $b\neq0,$ and $\left\vert a\right\vert \neq\left\vert b\right\vert ^{1/p}$. Solutions include we...

For a function from the unit interval to itself with constant slope and one discontinuity, the itineraries of the point of discontinuity are called the critical itineraries. These critical itineraries play a significant role in the study of β-expansions (with positive or negative β) and fractal transformations. A combinatorial characterization of t...

Theorems and explicit examples are used to show how transformations between self-similar sets (general sense) may be continuous almost everywhere with respect to stationary measures on the sets and may be used to carry well known flows and spectral analysis over from familiar settings to new ones. The focus of this work is on a number of surprising...

Necessary and sufficient conditions for the symbolic dynamics of a Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.

A simple, yet unifying method is provided for the construction of tilings by tiles obtained from the attractor of an iterated function system (IFS). Many examples appearing in the literature in ad hoc ways, as well as new examples, can be constructed by this method. These tilings can be used to extend a fractal transformation defined on the attract...

A simple, yet unifying method is provided for the construction of tilings by tiles obtained from the attractor of an iterated function system (IFS). Many examples appearing in the literature in ad hoc ways, as well as new examples, can be constructed by this method. These tilings can be used to extend a fractal transformation defined on the attract...

We define and exemplify the continuations and the fast basin of an attractor of an IFS. Then we extend the standard symbolic IFS theory, concerning the dynamics of a contractive IFS on its attractor, to a symbolic description of the dynamics of an invertible IFS on a set that contains the fast basin of a point-fibred attractor. We use this descript...

Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Möbius IFS...

Iterated function systems have been most extensively studied when the functions are affine transformations of Euclidean space and, more recently, projective transformations on real projective space. This paper investigates iterated function systems consisting of Möbius transformations on the extended complex plane or, equivalently, on the Riemann s...

Motivated by the question of how macromolecules assemble,the notion of an 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 asse...

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.

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 fo...

Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in simple situations. This paper concerns binary radix systems, i.e., positional number systems with digits 0 and 1. Our combinatorial approach is to construct infinitely many binary radix systems, each one from a sing...

A uniform function of the unit interval to itself is a piecewise continuous function with a single point of discontinuity and with two linear branches of the same slope. The itineraries of the point of discontinuity, which arise in the study of fractal transformations, are called the critical itineraries. A combinatorial characterization of the cri...

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...

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 transfor...

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 homeomorp...

We develop the theory of fractal homeomorphisms generated from pairs of overlapping affine iterated function systems.

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 repre...

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. C...

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 se...

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...

Let T be a tree all of whose internal vertices have degree at least three. In 1983 Jamison conjectured in JCT B that the average order of a subtree of T is at least half the order of T. In this paper a proof is provided. In addition, it is proved that the average order of a subtree of T is at most three quarters the order of T. Several open questio...

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...

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 repre...

This paper investigates a multi-resolution digital Earth model called PYXIS, which was developed by PYXIS Innovation Inc. The PYXIS hexagonal grids employ an efficient hierarchical labeling scheme for addressing pixels. We provide a recursive definition of the PYXIS grids, a systematic approach to the labeling, an algorithm to add PYXIS labels, and...

Let Δ be an n-dimensional simplex in Euclidean space \mathbbEn{\mathbb{E}}^n contained in an n-dimensional closed ball B. The following question is considered. Given any point x Î \trianglex \,{\in}\,\triangle, does there exist a reflection r : \mathbbEn ® \mathbbEnr : {\mathbb{E}}^n \rightarrow {\mathbb{E}}^n in one of the facets of Δ such t...

Let C(T) denote the poset of subtrees of a tree T with respect to the inclusion ordering. Jacobson, K�zdy and Seif gave a single example of a tree T for which C(T) is not Sperner, answering a question posed by Penrice. The authors then ask whether there exist an infinite family of trees T such that C(T) is not Sperner. This paper provides such a fa...

The computation of the Discrete Fourier Transform for a general lattice in ℝ d can be reduced to the computation of the standard 1-dimensional Discrete Fourier Transform. We provide a mathematically rigorous but simple treatment of this procedure and apply it to the DFT on the hexagonal lattice.

The problem of finding an approximation to a geometric line by a discrete line using pixels is ubiquitous in computer graphics applications. We show that this discrete line problem in ℝ n+1 , for grids of any shape, is equivalent to a geometry problem in ℝ n concerning the minimization of the distance that a certain type of closed polygonal path wa...

Over the past decade there has been interest in the computer representation of global data based on multi-resolution subdivisions of regular polyhedra. A simple and efficient indexing of the cells of such a subdivision, called A3-coordinates, is introduced. These can be used to encode the 4 Æ 3n + 2 cells at the nth level of resolution of the octah...

This paper introduces parity versions of familiar graph theoretic results, in particular results related to 2-connectedness. The even and odd circuit connected graphs are characterized. The realizable, even-realizable, alternating-realizable, dual realizable and dual even-realizable graphs are classified.

A generalization of the chromatic number of a graph is introduced such that the colors are integers modulo n, and the colors on adjacent vertices are required to be as far apart as possible.

The concept of strongly balanced graph is introduced. It is shown that there exists a strongly balanced graph with v vertices and e edges if and only if I ν – 1 ⩽ e ⩽(). This result is applied to a classic question of Erdös and Rényi: What is the probability that a random graph on n vertices contains a given graph? A rooted version of this problem...

A graph is called locally homogeneous if the subgraphs induced at any two points are isomorphic. in this Note we give a method for constructing locally homogeneous graphs from groups. the graphs constructable in this way are exactly the locally homogeneous graphs with a point symmetric universal cover. As an example we characterize the graphs that...

This paper introduces parity versions of familiar graph theoretic results, in par- ticular results related to 2-connectedness. The even and odd circuit connected graphs are characterized. The realizable, even-realizable, alternating-realizable, dual realizable and dual even-realizable graphs are classied.

Integrity, a measure of network reliability, is defined as I(G)=min S⊂V {|S|+m(G-S)}, where G is a graph with vertex set V and m(G-S) denotes the order of the largest component of G-S. We prove an upper bound of the following form on the integrity of any cubic graph with n vertices: I(G)<1 3n+O(n)· Moreover, there exist an infinite family of connec...

For a graph G, a graph recurrence sequence of vectors is defined by the recurrencewhere A is the adjacency matrix of G and is an initial vector. Each vector in this sequence can be thought of as a vertex labeling of G, the label at a given vertex at step t+1 obtained by summing the values at the adjacent vertices at step t. Based on graphical seque...

A set U of vertices of a graph G is called a geodetic set if the union of all the geodesics joining pairs of points of U is the whole graph G. One result in this paper is a tight lower bound on the minimum number of vertices in a geodetic set. In order to obtain that result, the following extremal set problem is solved. Find the minimum cardinality...

The concepts of separation index of a graph and of a surface are introduced. We prove that the separation index of the sphere is 3. Also the separation index of any graph faithfully embedded in a surface of genus g is bounded by a funtion of g. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 53–61, 2002

Perhaps the best known algorithm in combinatorial optimization is the greedy algorithm. A natural question is for which optimization problems does the greedy algorithm produce an optimal solution? In a sense this question is answered by a classical theorem in matroid theory due to Rado and Edmonds. In the matroid case, the greedy algorithm solves t...

The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are the special case where W is the symmetric group (the A n case) and P is a m...

A Coxeter matroid is a generalization of matroid, ordinary matroid being the case corresponding to the family of Coxeter groups A n , which are isomorphic to the symmetric groups. A basic result in the subject is a geometric characterization of Coxeter matroid in terms of the matroid polytope, a result first stated by Gelfand and Serganova. This pa...

This is an expository paper on digit tiling of Euclidean space, a special kind of self-affine tiling by translates of a single tile. In particular, the following topics are discussed: the construction of digit tiles and the construction of the boundary, the Hausdorff dimension of the boundary, the relation between digit tiles and positional number...

This paper considers representations of graphs as rectangle-visibility graphs and as doubly linear graphs. These are, respectively, graphs whose vertices are isothetic rectangles in the plane with adjacency determined by horizontal and vertical visibility, and graphs that can be drawn as the union of two straight-edged planar graphs. We prove that...

A polynomial time algorithm is given for deciding, for a given polyomino P , whether there exists an isohedral tiling of the Euclidean plane by isometric copies of P . The decidability question for general tilings by copies of a single polyomino, or even periodic tilings by copies of a single polyomino, remains open.

. An eective method is given for computing the Hausdor dimension of the boundary of a self-similar digit tile T in n-dimensional Euclidean space: dim H (@T ) = log log c ; where 1=c is the contraction factor and is the largest eigenvalue of a certain contact matrix rst dened by Grochenig and Haas. 1. Introduction In the book, Classics on Fractals...

The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are a special case, the case W = A (isomorphic to the symmetric group Sym_n+1)...

. Coxeter matroids, introduced by I.M. Gelfand and V. Serganova, are combinatorial structures associated with any nite Coxeter group and its parabolic subgroup; they include ordinary matroids as a special case. A basic result in the subject is a geometric characterization of Coxeter matroid, rst stated by Gelfand and Serganova. This paper presents...

An effective method is given for computing the Hausdorff dimension of the boundary of a self-similar digit tile T in n-dimensional Euclidean space: where 1/c is the contraction factor and λ is the largest eigenvalue of a certain contact matrix first defined by Gröchenig and Haas.

All self-replicating lattice tilings of the plane can be constructed using special iterated function systems (IFS). Certain self-replicating curves can be constructed using the recurrent set method (RS). A bijection between the IFS parameters and the RS parameters is such that the curve K produced by the RS parameters is the boundary of the tile T...

The non-revisiting path conjecture for polytopes, which is equivalent to the Hirsch conjecture, is open. However, for polyhedral maps on surfaces, we have recently proved the conjecture false for all orientable surfaces of genus g/> 2 and all nonorientable surfaces of nonorientable genus h/>4. In this paper, a unified, elementary proof of the non-r...

Several problems,concerning,the distribution of cycle lengths in a graph have been,proposed,by P. Erd¨ os and colleagues. In this note two variations of the following such question,are answered.,In a simple,graph where,every vertex has degree,at least three, must there exist two cycles whose lengths differ by one or two? c 1998 John Wiley & Sons, I...

Introduction. An appealing aspect of matroid theory is the multiple approaches to the subject. A matroid can be defined in terms of independent sets, bases, circuits, closure, rank, lattice of flats, etc. In this paper we concentrate on a less familiar characterization, but one deserving of more attention. This is the approach by way of the matroid...

A famous theorem of Ryser asserts that a v × v zero-one matrix A satisfying AAT = (k − λ)I + λJ with k ≠ λ must satisfy k + (v − 1) λ = k2 and ATA = (k − λ)I + λJ; such a matrix A is called the incidence matrix of a symmetric block design. We present a new, elementary proof of Ryser's theorem and give a characterization of the incidence matrices of...

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say...

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The nonrevisiting path conjecture for polytopes, which is equivalent to the Hirsch conjecture, is open. However, for surfaces, the nonrevisiting path conjecture is known to be true for polyhedral maps on the sphere, projective plane, torus, and a Klein bottle. Barnette has provided counterexamples on the orientable surface of genus 8 and nonorienta...

It is proved that the set of branches of a graphG is reconstructible except in a very special case. More precisely the set of branches of a graphG is reconstructible unless all the following hold: (1) the pruned center ofG is a vertex or an edge, (2)G has exactly two branches, (3) one branch contains all the vertices of degree one ofG and the other...

Arep-tiling is a self replicating, lattice tiling ofR n .Lattice tiling means a tiling by translates of a single compact tile by the points of a lattice, andself-replicating means that there is a non-singular linear map: R n Rn such that, for eachT , the image(T) is, in turn, tiled by . This topic has recently come under investigation, not only b...

A map is an embedding of a graph into a surface so that each face is simply connected. Geometric duality, whereby vertices and faces are reversed, is a classic construction for maps. A generalization of map duality is given and discussed both graph and group theoretically.

The classical lattice A∗n, whose Voronoi cells tile Euclidean n-space by permutohedra, can be given the generalized balance ternary ring structure GBTn in a natural way as a quotient ring of [x]. The ring GBTn can also be considered as the set of all finite sequences s0s1…sk, with si ∈ GBTn⧸αGBTn for all i, where α is an appropriately chosen elemen...

A theory of replicating tessellation of ℝ n is developed that simultaneously generalizes radix representation of integers and hexagonal addressing in computer science. The tiling aggregates tesselate Euclidean space so that the (m+1)st aggregate is, in turn, tiled by translates of the mth aggregate, for each m in exactly the same way. This induces...

For n sufficiently large the order of a smallest balanced extension of a graph of order n is, in the worst case, ⌊(n + 3)2/8⌋. © 1993 John Wiley & Sons, Inc.