A Note on Sarkovskiĭ's Theorem in Connected Linearly Ordered Spaces

2003 ◽  
Vol 13 (07) ◽  
pp. 1665-1671 ◽  
Author(s):  
D. Alcaraz ◽  
M. Sanchis
Keyword(s):  
Cardinal Number ◽  
Countable Subset ◽  
Minimal Sets ◽  
Compact Subspace ◽  
Countable Space ◽  
Ordered Space

We prove that, for a connected linearly ordered space L, the following conditions are equivalent: (1) L satisfies Sarkovskiĭ's Theorem, (2) there exist turbulent functions on L, and (3) there exists a compact subspace of L which satisfies Sarkovskiĭ's Theorem. Our results are applied in two ways. Firstly, we show that there exist connected linearly ordered spaces without infinite minimal sets; secondly, for each cardinal number λ of uncountable cofinality, we construct a connected linearly ordered space L such that: (1) L is a compact nonfirst countable space satisfying Sarkovskiĭ's Theorem, (2) L admits a dense first countable subset, and (3) the density of L is λ.

Characterization of Certain Binary Relations on Connected Ordered Spaces

1963 ◽  
Vol 15 ◽  
pp. 397-411
Author(s):  
James E. L'Heureux
Keyword(s):  
Natural Setting ◽  
Binary Relations ◽  
Identity Function ◽  
Minimal Sets ◽  
End Point ◽  
Ordered Space

In an earlier paper (2) reflexive transitive binary relations were considered on a connected ordered space. These relations were topologically restricted and their minimal sets were either an end point of the space or empty. It was shown that these relations could be characterized as one of the two orders of the space. Viewing the situation somewhat differently as suggested by I. S. Krule, one could say that this class of relations was characterized in terms of the identity function on the space. In this case the relations are considered in their natural setting, the product of the space with itself.

scholarly journals The partially pre-ordered set of compactifications of Cp(X, Y)

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
A. Dorantes-Aldama ◽  
R. Rojas-Hernández ◽  
Á. Tamariz-Mascarúa
Keyword(s):  
Ordered Set ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Countable Space ◽  
Generalized Ordered Space ◽  
Ordered Space ◽  
Countable Character ◽  
Corson Compact ◽  
First Countable Space ◽  
Isolated Point

AbstractIn the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where Cp(X, Y) is the space of continuous functions from X to Y with the topology inherited from the Tychonoff product space YX. We write Cp(X) instead of Cp(X, R).We prove that for a first countable space Y, K(Cp(X, Y)) is not a lattice if any of the following cases happen:(a) Y is not locally compact,(b) X has only one non isolated point and Y is not compact.Furthermore, K(Cp(X)) is not a lattice when X satisfies one of the following properties:(i) X has a non-isolated point with countable character,(ii) X is not pseudocompact,(iii) X is infinite, pseudocompact and Cp(X) is normal,(iv) X is an infinite generalized ordered space.Moreover, K(Cp(X)) is not a lattice when X is an infinite Corson compact space, and for every space X, K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.

Reconstruction of the Ancient Numeral System in Basque Language

Philology ◽  
2019 ◽  
Vol 4 (2018) ◽  
pp. 157-172
Author(s):  
FERNANDO GOMEZ-ACEDO ◽  
ENEKO GOMEZ-ACEDO
Keyword(s):  
Cardinal Number ◽  
Counting System ◽  
Cardinal Numbers ◽  
Original Meaning ◽  
Diachronic Development ◽  
Insight Into

Abstract In this work a new insight into the reconstruction of the original forms of the first Basque cardinal numbers is presented and the identified original meaning of the names given to the numbers is shown. The method used is the internal reconstruction, using for the etymologies words that existed and still exist in Basque and other words reconstructed from the proto-Basque. As a result of this work it has been discovered that initially the numbers received their name according to a specific and logic procedure. According to this ancient method of designation, each cardinal number received its name based on the hand sign used to represent it, thus describing the position adopted by the fingers of the hand to represent each number. Finally, the different stages of numerical formation are shown, which demonstrate a long and diachronic development of the whole counting system.

scholarly journals Minimal F2-flows

1985 ◽  
Vol 39 (2) ◽  
pp. 187-193
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Affine Transformations ◽  
Dimensional Torus ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet σ be an ergodic endomorphism of the r–dimensional torus and Π a semigroup generated by two affine transformations lying above σ. We show that the flow defined by Π admits minimal sets of positive Hausdorff dimension and we give necessary and sufficient conditions for this flow to be minimal.

Invariant Radon measures and minimal sets for subgroups of Homeo+(R)

2020 ◽  
Vol 285 ◽  
pp. 107378
Author(s):  
Hui Xu ◽  
Enhui Shi ◽  
Yiruo Wang
Keyword(s):  
Radon Measures ◽  
Minimal Sets

Retraction of a Compact Subspace onto Its Boundary

2003 ◽  
Vol 139 (2) ◽  
pp. 169-172
Author(s):  
M. A. S�nchez-Granero
Keyword(s):  
Compact Subspace

IMPRECISE CLASSIFICATION WITH CREDAL DECISION TREES

2012 ◽  
Vol 20 (05) ◽  
pp. 763-787 ◽  
Author(s):  
JOAQUÍN ABELLÁN ◽  
ANDRÉS R. MASEGOSA
Keyword(s):  
Decision Trees ◽  
Naive Bayes ◽  
Cardinal Number ◽  
Naïve Bayes ◽  
Classification Method ◽  
Cost Matrix ◽  
Imprecise Dirichlet Model ◽  
Dirichlet Model

In this paper, we present the following contributions: (i) an adaptation of a precise classifier to work on imprecise classification for cost-sensitive problems; (ii) a new measure to check the performance of an imprecise classifier. The imprecise classifier is based on a method to build simple decision trees that we have modified for imprecise classification. It uses the Imprecise Dirichlet Model (IDM) to represent information, with the upper entropy as a tool for splitting. Our new measure to compare imprecise classifiers takes errors into account. Thus far, this has not been considered by other measures for classifiers of this type. This measure penalizes wrong predictions using a cost matrix of the errors, given by an expert; and it quantifies the success of an imprecise classifier based on the cardinal number of the set of non-dominated states returned. To compare the performance of our imprecise classification method and the new measure, we have used a second imprecise classifier known as Naive Credal Classifier (NCC) which is a variation of the classic Naive Bayes using the IDM; and a known measure for imprecise classification.

Uncountable models and infinitary elementary extensions

1973 ◽  
Vol 38 (3) ◽  
pp. 460-470 ◽  
Author(s):  
John Gregory
Keyword(s):  
Countable Subset ◽  
Elementary Extension ◽  
Admissible Set ◽  
Natural Numbers ◽  
Elementary Equivalence ◽  
Countable Structure

Let A be a countable admissible set (as defined in [1], [3]). The language LA consists of all infinitary finite-quantifier formulas (identified with sets, as in [1]) that are elements of A. Notationally, LA = A ∩ Lω1ω. Then LA is a countable subset of Lω1ω, the language of all infinitary finite-quantifier formulas with all conjunctions countable. The set is the set of Lω1ω sentences defined in 2.2 below. The following theorem characterizes those A-Σ1 sets Φ of LA sentences that have uncountable models.Main Theorem (3.1.). If Φ is an A-Σ1set of LA sentences, then the following are equivalent:(a) Φ has an uncountable model,(b) Φ has a model with a proper LA-elementary extension,(c) for every , ⋀Φ → C is not valid.This theorem was announced in [2] and is proved in §§3, 4, 5. Makkai's earlier [4, Theorem 1] implies that, if Φ determines countable structure up to Lω1ω-elementary equivalence, then (a) is equivalent to (c′) for all , ⋀Φ → C is not valid.The requirement in 3.1 that Φ is A-Σ1 is essential when the set ω of all natural numbers is an element of A. For by the example of [2], then there is a set Φ LA sentences such that (b) holds and (a) fails; it is easier to show that, if ω ϵ A, there is a set Φ of LA sentences such that (c) holds and (b) fails.

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