James Keesling

An improved algorithm for computing topological entropy

A new algorithm is presented for computing the topological entropy of a unimodal map of the interval. The accuracy of the algorithm is discussed and some graphs of the topological entropy which are obtained using the algorithm are displayed.

Normality and properties related to compactness in hyperspaces

Introduction. Let X be a regular T1 topological space and 2X the space of all closed nonempty subsets of X with the finite topology [8, Definition 1.7]. In [6] Ivanova has shown that if X is a noncompact ordinal space, then 2X is nonnormal. In this paper we give a new proof of this fact. This result is then used to show that several properties of 2X are equivalent to the compactness of X. It is not known if the normality of 2X is equivalent to the compactness of X. There are some partial results in this direction though. The paracompactness of 2x is shown to be equivalent to the compactness of X and the normality of 22X is also shown to be equivalent to the compactness of X. In the last part of the paper some properties related to the countable compactness of 2x are investigated. Notation. Because of our assumptions on X, X= { {x} :xX} is a closed subset of 2X homeomorphic to X. The set 5F,(X) = { FCX: F has at most n points is also closed. Furthermore, the space 2X is Hausdorff. For notation and further basic results on hyperspaces see [7] or [8]. In particular we use (U1, U l *, Un)= {A 2x:ACU A U; and A r Ui # 0 for all i }. If each Ui is open in X, then ( U1, * * , U,,) is open in 2X and the set of such sets in 2X forms a basis for 2X. By considering such basic open sets it is clear that the set 5F(X) of finite subsets of X is dense in 2X. We denote the cardinality of a set Z by I zi .

The Hausdorff Dimension of the Boundary of a Self-Similar Tile

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: formula here where 1/c is the contraction factor and λ is the largest eigenvalue of a certain contact matrix first defined by Grochenig and Haas.

Asymptomatic spread of huanglongbing and implications for disease control

Significance Huanglongbing (HLB) is a vector-transmitted bacterial infection of citrus trees that poses a major threat to the citrus industry in Florida, Texas, and California. Current control strategies that focus on the vector, the Asian citrus psyllid Diaphorina citri, are usually initiated when the trees become symptomatic, anywhere from 10 mo to several years after initial infection. We show, experimentally, that newly infected young leaves can become infectious within 10–15 d after receiving an inoculum of bacteria from an adult psyllid. We then show by microsimulation of the asymptomatic spread of HLB through a grove under different invasion scenarios and control strategies that reduction of up to 75% of adult psyllids and nymphs can enhance citrus production. Huanglongbing (HLB) is a bacterial infection of citrus trees transmitted by the Asian citrus psyllid Diaphorina citri. Mitigation of HLB has focused on spraying of insecticides to reduce the psyllid population and removal of trees when they first show symptoms of the disease. These interventions have been only marginally effective, because symptoms of HLB do not appear on leaves for months to years after initial infection. Limited knowledge about disease spread during the asymptomatic phase is exemplified by the heretofore unknown length of time from initial infection of newly developing cluster of young leaves, called flush, by adult psyllids until the flush become infectious. We present experimental evidence showing that young flush become infectious within 15 d after receiving an inoculum of Candidatus Liberibacter asiaticus (bacteria). Using this critical fact, we specify a microsimulation model of asymptomatic disease spread and intensity in a grove of citrus trees. We apply a range of psyllid introduction scenarios to show that entire groves can become infected with up to 12,000 psyllids per tree in less than 1 y, before most of the trees show any symptoms. We also show that intervention strategies that reduce the psyllid population by 75% during the flushing periods can delay infection of a full grove, and thereby reduce the amount of insecticide used throughout a year. This result implies that psyllid surveillance and control, using a variety of recently available technologies, should be used from the initial detection of invasion and throughout the asymptomatic period.

Computing the topological entropy of maps of the interval with three monotone pieces

An algorithm is presented for computing the topological entropy of a piecewise monotone map of the interval having three monotone pieces. The accuracy of the algorithm is discussed and some graphs of the topological entropy obtained using the algorithm are displayed. Some of the ideas behind the algorithm have application to piecewise monotone functions with more than three monotone pieces.

The group of homeomorphisms of a solenoid

Let X be a topological space. An n-mean on X is a continuous function ,u: XX which is symmetric and idempotent. In the first part of this paper it is shown that if X is a compact connected abelian topological group, then X admits an n-mean if and only if H1(X, Z) is n-divisible where Hm(X, Z) is m-dimensional Cech cohomology with integers Z as coefficient group. This result is used to show that if Ia is a solenoid and Aut(la) is the group of topological group automorphisms of Ya, then Aut( (a) is algebraically Z2 x G where G is 1to, Zn, or D 1 Z. For a given Y., the structure of Aut(Ia) is determined by the n-means which a admits. Topologically, Aut(la) is a discrete space which has two points or is countably infinite. The main result of the paper gives the precise topological structure of the group of homeomorphisms G(a) of a solenoid Ya with the compact open topology. In the last section of the paper it is shown that G(a) is homeomorphic to Yax 12 x Aut(Xa) where 12 is separable infinite-dimensional Hilbert space. The proof of this result uses recent results in infinite-dimensional topology and some techniques using flows developed by the author in a previous paper. Introduction. Let X be a topological space. An n-mean for n > 2 on X is a continuous function 1i: Xn -X having the property that yu(x, * A., n) t(x T(l) *9 **, xr(n)) for any permutation 7n of II, * **, nI and p(x, x, * * *, x) = x for all x in X. We say simply that ti is symmetric and idempotent, respectively. Aumann showed that the circle T does not admit an n-mean for any n in [1]. In Eckmann [31 and Eckmann, Ganea, and Hilton [4] this result was extended to show that many other spaces do not support n-means. In particular, in [4] it is shown that if X is a compact connected polyhedron and X admits an n-mean for some n, then X is contractible. For the most part the above authors have devoted their efforts to widening the class of spaces known not to admit an n-mean. In the first section of this paper we show that a large class of compact connected abelian groups admit n-means for various ns. We give necessary and sufficient conditions that a compact connected abelian topological group H admit an n-mean. Among the equivalent conditions we show that H admits an n-mean if and only if Presented to the Society, January 19, 1972; received by the editors September 28, 1971. AMS 1970 subject classifications. Primary 57E05, 22B05.

Embedding Tn-like continua in Euclidean space

Abstract Many authors have been concerned with embedding ∏-like continua in Rn where ∏ is some collection of polyhedra or manifolds. A similar concern has been embedding ∏-like continua in Rn up to shape. In this paper we prove two main theorems. Theorem: If n ⩾ 2 and X is Tn-like, then X embeds in R2n. This result was conjectured by McCord for the case H1(X) finitely generated and proved by McCord for the case that H1(X) = 0 using a theorem of Isbell. The second theorem is a shape embedding theorem. Theorem: If X is Tn-like, then X embeds in Rn+2 up to shape. This theorem is proved by showing that an n-dimensional compact connected abelian topological group embeds in Rn+2. Any Tn-like continuum is shape equivalent to a k-dimensional compact connected abelian topological group for some 0 ⩽ k ⩽ n.

Strange adding machines

We show that given an adding machine of type

An algebraic property of the Čech cohomology groups which prevents local connectivity and movability

\alpha

A flexible simulation platform to quantify and manage emergency department crowding.

, for a dense set of parameters