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).

scholarly journals A characterization of locally connected unicoherent continua

1969 ◽  
Vol 10 (3-4) ◽  
pp. 257-265
Author(s):  
Philip Bacon
Keyword(s):  
Metric Space ◽  
Finite Sequence ◽  
Compact Metric Space

If ε > 0, a subset M of a metric space is said to be ε-connected if for each pair p, q ∈ M there is a finite sequence a0, …, an such that each ai ∈ M, a0 = ρ an = q and the distance from ai−1 to ai is less than ε whenever 0 < i ≦n. It is known [1, p. 117, Satz 1] that a compact metric space is connected if and only if for each ε > 0 it is ε-connected. We present here a proof of an analogous characterization of locally connected unicoherent compacta.

Martin's axiom and Hausdorif measures

1974 ◽  
Vol 75 (2) ◽  
pp. 193-197 ◽  
Author(s):  
A. J. Ostaszewski
Keyword(s):  
Metric Space ◽  
Measure Zero ◽  
Continuum Hypothesis ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
The Continuum

AbstractA theorem of Besicovitch, namely that, assuming the continuum hypothesis, there exists in any uncountable complete separable metric space a set of cardinality the continuum all of whose Hausdorif h-measures are zero, is here deduced by appeal to Martin's Axiom. It is also shown that for measures λ of Hausdorff type the union of fewer than 2ℵ0 sets of λ-measure zero is also of λ-measure zero; furthermore, the union of fewer than 2ℵ0 λ-measurable sets is λ-measurable.

Linear Chain of Coupled Fractional Oscillators: Response Dynamics and Its Continuum Limit

2007 ◽  
Author(s):  
B. N. Narahari Achar ◽  
Tanya Prozny ◽  
John W. Hanneken
Keyword(s):  
Continuum Limit ◽  
Equations Of Motion ◽  
Linear Chain ◽  
Harmonic Oscillators ◽  
Fractional Dynamics ◽  
Response Dynamics ◽  
Numerical Application ◽  
Diffusion Wave ◽  
A Chain ◽  
The Continuum

The standard model of a chain of simple harmonic oscillators of Condensed Matter Physics is generalized to a model of linear chain of coupled fractional oscillators in fractional dynamics. The set of integral equations of motion pertaining to the chain of harmonic oscillators is generalized by taking the integrals to be of arbitrary order according to the methods of fractional calculus to yield the equations of motion of a chain of coupled fractional oscillators. The solution is obtained by using Laplace transforms. The continuum limit of the equations is shown to yield the fractional diffusion-wave equation in one dimension. The solution and numerical application of the set of equations and the continuum limit there of are discussed.

Embedding Circle-Like Continua in the Plane

1962 ◽  
Vol 14 ◽  
pp. 113-128 ◽  
Author(s):  
R. H. Bing
Keyword(s):  
Linear Chain ◽  
Simple Closed Curve ◽  
Finite Sequence ◽  
Closed Curve ◽  
Mesh Less

A finite sequence of open sets L1 L2, … , Ln is called a linear chain if each Li intersects only the L's adjacent to it in the sequence. The finite sequence is a circular chain if we also insist that the first and last links intersect each other. The 1-skeleton of the covering is an arc for a linear chain and a simple closed curve for a linear chain.A compact metric continuum X is called snake-like if for each ∈ > 0, X can be covered by a linear chain of mesh less than ∈. Likewise, X is called circle-like if for each ∈ > 0, X can be irreducibly covered with a circular chain of mesh less than ∈. This definition is more restrictive than that given in (3, p. 210) for there a pseudo-arc is not circle-like but here it is. The present usage is in keeping with definitions of Burgess.

scholarly journals THE CONTINUUM LIMIT OF DISCRETE GEOMETRIES

2006 ◽  
Vol 03 (02) ◽  
pp. 285-313 ◽  
Author(s):  
MANFRED REQUARDT
Keyword(s):  
Metric Space ◽  
Continuum Limit ◽  
Metric Spaces ◽  
Scaling Limit ◽  
Space Time ◽  
Pure Mathematics ◽  
Modern Physics ◽  
Core Concepts ◽  
Scaling Dimension ◽  
The Continuum

In various areas of modern physics and in particular in quantum gravity or foundational space–time physics, it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be constructed from a more primordial and basically discrete underlying substratum, which may behave in a quite erratic and irregular way. We develop such a framework within the category of general metric spaces by combining recent work of our own and ingeneous ideas of Gromov et al. developed in pure mathematics. A central role is played by two core concepts. For one, the notion of intrinsic scaling dimension of a (discrete) space or, in mathematical terms, the growth degree of a metric space at infinity, on the other hand, the concept of a metrical distance between general metric spaces and an appropriate scaling limit (called by us a geometric renormalization group) performed in this metric space of spaces. In doing this, we prove a variety of physically interesting results about the nature of this limit process, properties of the limit space, e.g., what preconditions qualify it as a smooth classical space–time and, in particular, its dimension.

Response Dynamics in the Continuum Limit of the Lattice Dynamical Theory of Viscoelasticity (Fractional Calculus Approach)

2009 ◽  
Author(s):  
B. N. Narahari Achar ◽  
John W. Hanneken
Keyword(s):  
Integral Equations ◽  
Continuum Limit ◽  
Equations Of Motion ◽  
Linear Chain ◽  
Dynamical Theory ◽  
Harmonic Oscillators ◽  
Fractional Integrals ◽  
Response Dynamics ◽  
Dynamical Equations ◽  
The Continuum

A fractional diffusion-wave equation is derived in the continuum limit of the lattice dynamical equations of motion of a chain of coupled fractional oscillators obtained from the integral equations of motion of a linear chain of simple harmonic oscillators by generalization of the ordinary integrals into ones involving fractional integrals. The set of integral equations of motion pertaining to the chain of coupled fractional oscillators in the continuum limit is solved by using Laplace transforms. The response of the system to impulse and sinusoidal forcing is studied. Numerical applications are discussed with particular reference to energy flow and dissipation.

scholarly journals Fixed Points can Characterise Curves

1978 ◽  
Vol 21 (2) ◽  
pp. 207-211 ◽  
Author(s):  
Helga Schirmer
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Metric Space ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
The Given ◽  
The Continuum

AbstractThe concept of a firm fixed point of a selfmap of a metric space is introduced. Loosely speaking a fixed point is firm if it cannot be moved to a point nearby with the help of a map which is arbitrarily close to the given map. It is shown that a continuum always admits a selfmap with a firm fixed point if the continuum contains a triod and if the vertex of the triod has a neighbourhood which is a dendrite. This condition holds in particular for local dendrites. Hence a local dendrite is an arc or a simple closed curve if and only if it does not admit a selfmap which has a firm fixed point.

scholarly journals RISE OF THE CENTRIST: FROM BINARY TO CONTINUOUS OPINION DYNAMICS

2008 ◽  
Vol 19 (09) ◽  
pp. 1459-1475 ◽  
Author(s):  
GEORGE A. BAKER ◽  
JAMES P. HAGUE
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Continuum Limit ◽  
Linear Chain ◽  
Opinion Dynamics ◽  
Memory Effects ◽  
Time Step ◽  
Binary Model ◽  
Long Time ◽  
The Continuum

We propose a model that extends the binary "united we stand, divided we fall" opinion dynamics of Sznajd-Weron to handle continuous and multi-state discrete opinions on a linear chain. Disagreement dynamics are often ignored in continuous extensions of the binary rules, so we make the most symmetric continuum extension of the binary model that can treat the consequences of agreement (debate) and disagreement (confrontation) within a population of agents. We use the continuum extension as an opportunity to develop rules for persistence of opinion (memory). Rules governing the propagation of centrist views are also examined. Monte Carlo simulations are carried out. We find that both memory effects and the type of centrist significantly modify the variance of average opinions in the large timescale limits of the models. Finally, we describe the limit of applicability for Sznajd-Weron's model of binary opinions as the continuum limit is approached. By comparing Monte Carlo results and long time-step limits, we find that the opinion dynamics of binary models are significantly different to those where agents are permitted more than 3 opinions.

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