On the Characterization of the Set of Points of λ-Continuity

1923 ◽  
Vol 25 (2) ◽  
pp. 118
Author(s):  
Henry Blumberg
Keyword(s):  
Set Of Points

A characterization of sets of n points which determine n hyperplanes

1968 ◽  
Vol 64 (3) ◽  
pp. 585-588 ◽  
Author(s):  
J. G. Basterfield ◽  
L. M. Kelly
Keyword(s):  
Projective Space ◽  
The Other ◽  
Finite Projective Space ◽  
Other Hand ◽  
Finite Set ◽  
Set Of Points

Suppose N is a set of points of a d-dimensional incidence space S and {Ha}, a ∈ I, a set of hyperplanes of S such that Hi ∈ {Ha} if and only if Hi ∩ N spans Hi. N is then said to determine {Ha}. We are interested here in the case in which N is a finite set of n points in S and I = {1, 2,…, n}; that is to say when a set of n points determines precisely n hyperplanes. Such a situation occurs in E3, for example, when N spans E3 and is a subset of two (skew) lines, or in E2 if N spans the space and n − 1 of the points are on a line. On the other hand, the n points of a finite projective space determine precisely n hyperplanes so that the structure of a set of n points determining n hyperplanes is not at once transparent.

Dynamics of $(e^{z} - 1)/z$: the Julia set and bifurcation

1998 ◽  
Vol 18 (6) ◽  
pp. 1363-1383 ◽  
Author(s):  
G. P. KAPOOR ◽  
M. GURU PREM PRASAD
Keyword(s):  
Real Root ◽  
Julia Set ◽  
Dynamical Behaviour ◽  
Positive Real ◽  
Finite Function ◽  
Positive Real Root ◽  
Transcendental Functions ◽  
Set Of Points

We describe the dynamical behaviour of the entire transcendental non-critically finite function $f_\lambda (z) = \lambda(e^z - 1)/z$, $\lambda > 0$. Our main result is to obtain a computationally useful characterization of the Julia set of $f_\lambda (z)$ as the closure of the set of points with orbits escaping to infinity under iteration, which in turn is applied to the generation of the pictures of the Julia set of $f_\lambda (z)$. Such a characterization was hitherto known only for critically finite entire transcendental functions [11]. We find that bifurcation in the dynamics of $f_\lambda (z)$ occurs at $\lambda = \lambda^{*}$ ($\approx 0.64761$) where $\lambda^\ast = {(x^{*})}^{2} /({e}^{x^{*}} -1)$ and $x^{*}$ is the unique positive real root of the equation $e^{x}(2 -x ) -2 = 0$.

scholarly journals Simultaneous dense and non-dense orbits for toral diffeomorphisms

2016 ◽  
Vol 37 (4) ◽  
pp. 1308-1322 ◽  
Author(s):  
JIMMY TSENG
Keyword(s):  
Baire Category ◽  
Geometric Characterization ◽  
The Other ◽  
Baire Category Theorem ◽  
Open Set ◽  
Category Theorem ◽  
Dense Orbits ◽  
Set Of Points ◽  
Toral Automorphisms

We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and non-dense forward orbits under the other is a dense, uncountable set. The pair of maps can be non-commuting. We also show the same for pairs of $C^{2}$-Anosov diffeomorphisms on the $2$-torus. (The pairs must satisfy slight constraints.) Our main tools are the Baire category theorem and a geometric construction that allows us to give a geometric characterization of the fractal that is the set of points with forward orbits that miss a certain open set.

scholarly journals On Hausdorff compactifications of non-locally compact spaces

1979 ◽  
Vol 2 (3) ◽  
pp. 481-486
Author(s):  
James Hatzenbuhler ◽  
Don A. Mattson
Keyword(s):  
Hausdorff Space ◽  
Locally Compact ◽  
Completely Regular ◽  
Compact Spaces ◽  
Hausdorff Compactifications ◽  
Set Of Points ◽  
Locally Compact Spaces

LetXbe a completely regular, Hausdorff space and letRbe the set of points inXwhich do not possess compact neighborhoods. AssumeRis compact. IfXhas a compactification with a countable remainder, then so does the quotientX/R, and a countable compactificatlon ofX/Rimplies one forX−R. A characterization of whenX/Rhas a compactification with a countable remainder is obtained. Examples show that the above implications cannot be reversed.

scholarly journals A characterization of the uniform convergence points set of some convergent sequence of functions

2021 ◽  
Vol 71 (2) ◽  
pp. 423-428
Author(s):  
Olena Karlova
Keyword(s):  
Uniform Convergence ◽  
Convergent Sequence ◽  
Normal Space ◽  
Disjoint Sequence ◽  
Isolated Points ◽  
Perfectly Normal Space ◽  
Set Of Points ◽  
Sequence Of Functions ◽  
Perfectly Normal

Abstract We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if X is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and A ⊆ X, then A is the set of points of the uniform convergence for some convergent sequence (fn ) n∈ω of functions fn : X → ℝ if and only if A is Gδ -set which contains all isolated points of X. This result generalizes a theorem of Ján Borsík published in 2019.

Interface Mobilities for Characterization of Structure-Borne Sound Sources with Multi-Point Coupling

2012 ◽  
Vol 98 (3) ◽  
pp. 384-391 ◽  
Author(s):  
H. A. Bonhoff ◽  
A. Eslami
Keyword(s):  
Single Point ◽  
Point Of View ◽  
Source Characterization ◽  
Sound Sources ◽  
Fourier Decomposition ◽  
Interface Mobility ◽  
Contact Points ◽  
Simplified Calculation ◽  
Set Of Points

The concept of source descriptor and coupling function is commonly recognized to form a rigorous basis for structure-borne sound source characterization. While this concept initially is valid for the single-point case only, it can be extended to sources with multi-point coupling by including the interface mobility approach. By considering a continuous interface that passes all contact points, the velocities, forces and mobilities are series expanded into interface orders by means of a spatial Fourier decomposition. The use of a continuous formulation for the multi-point case, however, can be problematic from a practical point of view. This paper discusses a reformulation of the interface mobility approach for a simplified calculation and clarified interpretation of the interface orders. With a discrete Fourier series as a basis for the interface mobility approach, the interface is reduced to a set of points and the interface orders are shown to describe the interplay of the data at the contact points. A discrete formulation furthermore yields simplified equations and a strict upper bound for the number of orders that have to be included, thus enhancing the practicability of interface mobilities for source characterization.

scholarly journals ON THE WIDTH AND ROUNDNESS OF A SET OF POINTS IN THE PLANE

1999 ◽  
Vol 09 (01) ◽  
pp. 97-108 ◽  
Author(s):  
MICHIEL SMID ◽  
RAVI JANARDHAN
Keyword(s):  
Voronoi Diagram ◽  
Intersection Point ◽  
Rigorous Proof ◽  
Minimum Width ◽  
Point Set ◽  
Set Of Points

Let S be a set of points in the plane. The width (resp. roundness) of S is defined as the minimum width of any slab (resp. annulus) that contains all points of S. We give a new characterization of the width of a point set. Also, we give a rigorous proof of the fact that either the roundness of S is equal to the width of S, or the center of the minimum-width annulus is a vertex of the closest-point Voronoi diagram of S, the furthest-point Voronoi diagram of S, or an intersection point of these two diagrams. This proof corrects the characterization of roundness used extensively in the literature.

scholarly journals Subcommuting and comparable iterative roots of order preserving homeomorphisms

2019 ◽  
Vol 26 (1/2) ◽  
pp. 203-210
Author(s):  
Veerapazham Murugan ◽  
Murugan Suresh Kumar
Keyword(s):  
Continuous Functions ◽  
Iterative Roots ◽  
Points Of Coincidence ◽  
Set Of Points

It is known that the iterative roots of continuous functions are not necessarily unique, if it exist. In this note, by introducing the set of points of coincidence, we study the iterative roots of order preserving homeomorphisms. In particular, we prove a characterization of identical iterative roots of an order preserving homeomorphism using the points of coincidence of functions.

scholarly journals Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra

10.29007/pw5g ◽  
2018 ◽  
Author(s):  
Larry Moss ◽  
Jayampathy Ratnayake ◽  
Robert Rose
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Topological Spaces ◽  
Fractal Sets ◽  
Cauchy Completion ◽  
Sweeping Generalization ◽  
Initial Algebra ◽  
Finite Metric Spaces ◽  
Set Of Points

This paper is a contribution to the presentation of fractal sets in terms of final coalgebras.The first result on this topic was Freyd's Theorem: the unit interval [0,1] is the final coalgebra ofa certain functor on the category of bipointed sets. Leinster 2011 offersa sweeping generalization of this result. He is able to represent many of what would be intuitivelycalled "self-similar" spaces using (a) bimodules (also called profunctors or distributors),(b) an examination of non-degeneracy conditions on functors of various sorts; (c) a construction offinal coalgebras for the types of functors of interest using a notion of resolution. In addition to thecharacterization of fractals sets as sets, his seminal paper also characterizes them as topological spaces.Our major contribution is to suggest that in many cases of interest, point (c) above on resolutionsis not needed in the construction of final coalgebras. Instead, one may obtain a number of spaces ofinterest as the Cauchy completion of an initial algebra,and this initial algebra is the set of points in a colimit of an omega-sequence of finite metric spaces.This generalizes Hutchinson's 1981 characterization of fractal attractors asclosures of the orbits of the critical points. In addition to simplifying the overall machinery, it also presents a metric space which is ``computationally related'' to the overall fractal. For example, when applied to Freyd's construction, our method yields the metric space.of dyadic rational numbers in [0,1].Our second contribution is not completed at this time, but it is a set of results on \emph{metric space}characterizations of final coalgebras. This point was raised as an open issue in Hasuo, Jacobs, and Niqui 2010,and our interest in quotient metrics comes from their paper. So in terms of (a)--(c) above, our workdevelops (a) and (b) in metric settings while dropping (c).

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