A non-chainable plane continuum with span zero

2011 ◽  
Vol 211 (2) ◽  
pp. 149-174 ◽  
Author(s):  
L. C. Hoehn
Keyword(s):  
Plane Continuum

A Simple Closed Curve is the Only Homogeneous Bounded Plane Continuum that Contains an Arc

1960 ◽  
Vol 12 ◽  
pp. 209-230 ◽  
Author(s):  
R. H. Bing
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
History Of ◽  
Homogeneous Plane ◽  
Plane Continuum

One of the unsolved problems of plane topology is the following:Question. What are the homogeneous bounded plane continua?A search for the answer has been punctuated by some erroneous results. For a history of the problem see (6).The following examples of bounded homogeneous plane continua are known : a point; a simple closed curve; a pseudo arc (2, 12); and a circle of pseudo arcs (6). Are there others?The only one of the above examples that contains an arc is a simple closed curve. In this paper we show that there are no other such examples. We list some previous results that point in this direction. Mazurkiewicz showed (11) that the simple closed curve is the only non-degenerate homogeneous bounded plane continuum that is locally connected. Cohen showed (8) that the simple closed curve is the only homogeneous bounded plane continuum that contains a simple closed curve.

Concerning Non-Planar Circle-Like Continua

1967 ◽  
Vol 19 ◽  
pp. 242-250 ◽  
Author(s):  
W. T. Ingram
Keyword(s):  
Metric Space ◽  
Linear Chain ◽  
Finite Sequence ◽  
Continuous Image ◽  
Plane Continuum ◽  
The Continuum

In this paper it is proved that if a circle-like continuum M cannot be embedded in the plane, then M is not a continuous image of any plane continuum (Theorem 5).Suppose that (S, ρ) is a metric space. A finite sequence of domains L1, L2, … , Ln is called a linear chain provided Li intersects Lj if and only if |i — j| ⩽ 1. If, in addition, there is a positive number ∊ such that, for each i, the diameter of Li is less than ∊, then the linear chain is called a linear ∊-chain. If for each positive number ∊ the continuum M can be covered by a linear ∊-chain, then M is said to be chainable (or snake-like) (2).

A homogeneous indecomposable plane continuum

1948 ◽  
Vol 15 (3) ◽  
pp. 729-742 ◽  
Author(s):  
R. H. Bing
Keyword(s):  
Plane Continuum

scholarly journals A non-dense plane continuum

1931 ◽  
Vol 37 (10) ◽  
pp. 720-723
Author(s):  
J. H. Roberts
Keyword(s):  
Plane Continuum

INTERIOR COMPONENTS OF A TILE ASSOCIATED TO A QUADRATIC CANONICAL NUMBER SYSTEM — PART II

Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 385-397 ◽  
Author(s):  
JULIEN BERNAT ◽  
BENOÎT LORIDANT ◽  
JÖRG THUSWALDNER
Keyword(s):  
Fundamental Domain ◽  
Number System ◽  
The Other ◽  
Integer Part ◽  
Finite Representation ◽  
Similarity Transformations ◽  
Canonical Number System ◽  
System Part ◽  
Self Similar ◽  
Plane Continuum

Let [Formula: see text] be a root of the polynomial p(x) = x2 + 4x + 5. It is well-known that the pair (α, {0, 1, 2, 3, 4}) forms a canonical number system, i.e., that each γ ∈ ℤ[α] admits a finite representation of the shape γ = a0 + a1α + ⋯ + aℓαℓ with ai ∈ {0, 1, 2, 3, 4}. The set [Formula: see text] of points with integer part 0 in this number system [Formula: see text] is called the fundamental domain of this canonical number system. It is a plane continuum with nonempty interior which induces a tiling of ℂ. Moreover, it has a disconnected interior [Formula: see text]. In the first paper of this series we described the closures C0, C1, C2 and C3 of the four largest components of [Formula: see text] as attractors of graph-directed self-similar sets. Each of these four sets is a translation of C0. We conjectured that the closures of the other components are images of C0 by similarity transformations. In this article we prove this conjecture. Moreover, we provide a graph from which the suitable similarities can be read off.

A Comparative Study on the Beam and Continuum Finite Element Models for the Rail–Wheel Vibration

2019 ◽  
Vol 19 (07) ◽  
pp. 1950076 ◽  
Author(s):  
Q. Xie ◽  
Y. X. Zhou ◽  
Y. Zhan ◽  
K. Y. Sze ◽  
W. S. Han
Keyword(s):  
Finite Element ◽  
Numerical Models ◽  
Computational Cost ◽  
Field Tests ◽  
Maintenance Cost ◽  
Fe Model ◽  
Beam Model ◽  
Rolling Wheel ◽  
Wheel Vibration ◽  
Plane Continuum

The rail–wheel interaction can induce train and track vibrations and consequently lead to noise impact, passengers’ discomfort, high maintenance cost, etc. Due to the complexity of the rail–wheel interaction and the high cost of field tests, as well as the difficulties in data collection, numerical analyses have been widely resorted to for predicting the train and track vibrations, for which numerous numerical models have been developed. According to track modeling approaches, numerical models can be generally divided into two categories, i.e. beam models and continuum finite element (FE) models. In this paper, these two models are systematically compared and discussed. First, a typical beam model of Wu and Thompson [T. X. Wu and D. Thompson, On the parametric excitation of the wheel/track system, J. Sound Vib. 278(4) (2004) 725–747.] is introduced, based on which a modified model is then established. Secondly, a plane continuum FE model with high mesh quality is established, in which the transition mesh generation, contact treatment and element size determination are presented. Numerical tests are conducted to validate the proposed plane FE model. Finally, both the beam and the plane continuum FE models are examined through typical rail–wheel interaction examples, in which the linear response of the track as well as the rail–wheel vibrations under both a single rolling wheel and two rolling wheels are analyzed. The results show that most of the vibration trends obtained from the two models agree well with each other. Nevertheless, it is noteworthy that the continuum FE model has superiorities, especially for analyzing vibrations at higher frequencies. The present study can be of considerable help for designers and engineers in the railway industry to achieve the trade-off between the simulation demands and the computational cost.

scholarly journals Current-Mode Network Structures Dedicated for Simulation of Dynamical Systems with Plane Continuum of Equilibrium

2018 ◽  
Vol 27 (09) ◽  
pp. 1830004 ◽  
Author(s):  
Jiri Petrzela ◽  
Tomas Gotthans ◽  
Milan Guzan
Keyword(s):  
Dynamical Systems ◽  
Current Mode ◽  
Equilibrium Structure ◽  
Third Order ◽  
Active Devices ◽  
Real Behavior ◽  
Dependent Model ◽  
The Rich ◽  
Frequency Dependent Model ◽  
Plane Continuum

This review paper describes different lumped circuitry realizations of the chaotic dynamical systems having equilibrium degeneration into a plane object with topological dimension of the equilibrium structure equals one. This property has limited amount (but still increasing, especially recently) of third-order autonomous deterministic dynamical systems. Mathematical models are generalized into classes to design analog networks as universal as possible, capable of modeling the rich scale of associated dynamics including the so-called chaos. Reference state trajectories for the chaotic attractors are generated via numerical analysis. Since used active devices can be precisely approximated by using third-level frequency dependent model, it is believed that computer simulations are close enough to capture real behavior. These simulations are included to demonstrate the existence of chaotic motion.

Homogeneous Continua Which are Almost Chainable1

1961 ◽  
Vol 13 ◽  
pp. 519-528 ◽  
Author(s):  
C. E. Burgess
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
History Of ◽  
Homogeneous Plane ◽  
Plane Continuum

The only known examples of nondegenerate homogeneous plane continua are the simple closed curve, the circle of pseudo-arcs (6), and the pseudo-arc (1; 13). Another example, called the pseudo-circle, has been suggested by Bing (2), but it has not been proved to be homogeneous. (Definitions of some of these terms and a history of results on homogeneous plane continua can be found in (6).) Of the three known examples, the pseudo-arc is both linearly chainable and circularly chainable, and the simple closed curve and the circle of pseudo-arcs are circularly chainable but not linearly chainable. It is not known whether every homogeneous plane continuum is either linearly chainable or circularly chainable. Bing has shown that a homogeneous continuum is a pseudo-arc provided it is linearly chainable (4).

scholarly journals A remark on compact semigroups having certain decomposition spaces embeddable in the plane

1971 ◽  
Vol 4 (1) ◽  
pp. 137-139 ◽  
Author(s):  
R. P. Hunter ◽  
L. W. Anderson
Keyword(s):  
Decomposition Spaces ◽  
Connected Subgroup ◽  
Decomposition Space ◽  
Compact Semigroups ◽  
Plane Continuum

Let S be a compact connected semigroup. If the decomposition space of S, under the action of a compact connected subgroup at the identity, is a plane continuum then this decomposition is a congruence.

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