Uncountable Research Articles

Infinite algebraic subgroups of the real Cremona group

We give the classification of the maximal infinite algebraic subgroups of the real Cremona group of the plane up to conjugacy and present a parametrisation space of each conjugacy class. Moreover, we show that the real plane Cremona group is not generated by a countable union of its infinite algebraic subgroups.

Pluripolar Hulls and Convergence Sets

The pluripolar hull of a pluripolar set E in ℙn is the intersection of all complete pluripolar sets in ℙn that contain E. We prove that the pluripolar hull of each compact pluripolar set in ℙn is Fσ. The convergence set of a divergent formal power series f(z0, … , zn) is the set of all “directions” ξ ∈ ℙn along which f is convergent. We prove that the union of the pluripolar hulls of a countable collection of compact pluripolar sets in ℙn is the convergence set of some divergent series f. The convergence sets on Γ := {[1 : z : ψ (z)] : z ∈ ℂ} ⊂ ℂ2 ⊂ ℙ2, where ψ is a transcendental entire holomorphic function, are also studied and we obtain that a subset on Γ is a convergence set in ℙ2 if and only if it is a countable union of compact projectively convex sets, and hence the union of a countable collection of convergence sets on Γ is a convergence set.

EXPLORING PRE-SERVICE MATHEMATICS TEACHERS’ UNDERSTANDINGS OF COUNTABILITY AND INFINITY IN WEBQUEST BASED LEARNING ENVIRONMENT

Infinity concept is difficult to understand because of its nature. Even if the concept of infinity doesn’t directly take part in mathematics curriculum, it takes place on many topics such as lines, planes, sets, irrational numbers, limit, differentiation, and integration. The present study examined pre-service mathematics teachers’ understandings of infinity and countability concepts in WebQuest based learning environment. Twenty-nine pre-service teachers participated at the study. Data were collected through a questionnaire developed by the researchers and semi-structured interviews. Qualitative data were analyzed by using phenomenology research method. The findings of this study revealed that pre-service teachers generally defined infinite sets as the sets whose elements continue infinitely. They inclined to define countable sets as bounded sets, finite sets, and the sets with known elements. Whereas most of the students stated countable finite sets and countable infinite sets were existent, they also expressed uncountable finite sets and uncountable infinite sets were non-existent. They used natural numbers set as an example for the countable infinite sets. This research presented some implications for teacher education programme in the light of obtained findings. Article visualizations:

Stable intersection of affine Cantor sets defined by two expanding maps

It is shown that in the context of affine Cantor sets with two increasing maps, the arithmetic sum of both of its elements is a Cantor set otherwise, it is a closure of countable union of nontrivial intervals. Also, a new family of pairs of affine Cantor sets is introduced such that each element of it has stable intersection. At the end, pairs of affine Cantor sets are characterized such that the sum of elements of each pair is a closed interval.

On countably uniform closed-spaces

Let X be a topological space and Cc(X) be the functionally countable subalgbera of C(X). We call X to be a countably uniform closed-space, briefly, a CU C-space, if Cc(X) is closed under uniform convergence. We investigate that countably uniform closedness need not closed under finite intersection and infinite product. It is shown that if X is a countable union of quasi-components, then X is a CU C-space. We characterize Cc-embedding and also -embedding in CU C-spaces. A subset S of X is called Zc-embedded, if each Z ∈ Zc(S) is the restriction of a zero-set of Zc(X). It is observed that in a zero-dimensional CU C-space, each Lindelöf subspae is Zc-embedded. Moreover, it is shown that in CU C-spaces, each Lindelöf subspace is Cc-embedded if and only if it is c-completely separated from each zero-set, which is disjoint from it. Also in latter spaces, it is observed that for each S ⊆ X, Cc-embedding, -embedding and Zc-embedding coincide, when S belongs to Zc(X) or it is a c-pseudocompact space. Finally, when X is both a CU C-space and a CP-space, then each Zc-embedded subspace is Cc-embedded (-embedded) in X.

Completely symmetric resistance forms on the stretched Sierpiński gasket

The stretched Sierpiński gasket, SSGfor short, is the space obtained by replacing every branching point of the Sierpiński gasket by an interval. It has also been called the “deformed Sierpiński gasket” or “Hanoi attractor”. As a result, it is the closure of a countable union of intervals and one might expect that a diffusion on SSG is essentially a kind of gluing of the Brownian motions on the intervals. In fact, there have been several works in this direction. There still remains, however, “reminiscence” of the Sierpiński gasket in the geometric structure of SSG and the same should therefore be expected for diffusions. This paper shows that this is the case. In this work, we identify all the completely symmetric resistance forms on SSG. A completely symmetric resistance form is a resistance form whose restriction to every contractive copy of SSG in itself is invariant under all geometrical symmetries of the copy, which constitute the symmetry group of the triangle. We prove that completely symmetric resistance forms on SSG can be sums of the Dirichlet integrals on the intervals with some particular weights, or a linear combination of a resistance form of the former kind and the standard resistance form on the Sierpiński gasket.

Approximation by Entire Functions on a Countable Union of Real-Axis Segments. 4. Inverse Theorem

For more than a century, the constructive description of functional classes in terms of the possible rate of approximation of its functions by means of functions chosen from a certain set remains among the most important problems of approximation theory. It turns out that the nonuniformity of the approximation rate due between the points of the domain of the approximated function is substantial. For instance, it was only in the mid-1950s that it was possible to constructively describe Holder classes on the segment [–1; 1] in terms of the approximation by algebraic polynomials. For that particular case, the constructive description requires the approximation at neighborhoods of the segment endpoints to be essentially better than the one in a neighborhood of its midpoint. A possible approximation quality test is to find out whether the approximation rate provides a possibility to reconstruct the smoothness of the approximated function. Earlier, we investigated the approximation of classes of smooth functions on a countable union of segments on the real axis. In the present paper, we prove that the rate of the approximation by the entire exponential-type functions provides the possibility to reconstruct the smoothness of the approximated function, i.e., a constructive description of classes of smooth functions is possible in terms of the specified approximation method. In an earlier paper, that result is announced for Holder classes, but the construction of a certain function needed for the proof is omitted. In the present paper, we use another proof; it does not apply the specified function.

Toledo frente a Madrid en la conformación del español moderno: el sistema pronominal átono

Los estudios sobre el pronombre átono objeto en español señalan de manera casi unánime en las últimas décadas que el factor más importante para la selección de las formas le(s)/lo(s) para complemento directo es el tipo semántico del sustantivo de referencia (contable o incontable). Dentro de este sistema llamado referencial, se incluyen hoy las provincias de Madrid y Toledo. En este artículo examinamos los datos basados en documentos archivísticos inéditos de ambas provincias para 1) establecer si este sistema estaba vigente en la Edad Media, 2) señalar la extensión a partir del siglo XVI, 3) comprobar si el sistema leísta se extendía a todos los sociolectos y 4) precisar las diferencias geográficas internas del castellano central.

Semipolar Sets and Intrinsic Hausdorff Measure

Given a “Green function” G on a locally compact space X with countable base, a Borel set A in X is called G-semipolar, if there is no measure ν ≠ 0 supported by A such that $G\nu :=\int G(\cdot ,y)\,d\nu (y)$ is a continuous real function on X. Introducing an intrinsic Hausdorff measuremG using G-balls B(x, ρ) := {y ∈ X : G(x, y) > 1/ρ}, it is shown that every set A in X with $m_{G}(A)<\infty $ is contained in a G-semipolar Borel set. This is of interest, since G-semipolar sets are semipolar in the potential-theoretic sense (countable unions of totally thin sets, hit by a corresponding process at most countably many times), if G is a genuine Green function. The result has immediate consequences for classical potential theory, Riesz potentials and the heat equation (where it solves an open problem). More generally, it is applied to metric measure spaces (X, d, μ), where a continuous heat kernel with upper and lower bounds of the form t−α/βΦj(d(x,y)t− 1/β), j = 1, 2, is given. Then the intrinsic Hausdorff measure on X is equivalent to an ordinary Hausdorff measure mα−β. For the corresponding space-time structure on X × ℝ, the intrinsic Hausdorff measure turns out to be equivalent to an anisotropic Hausdorff measure mα,β.

Visibility graphs and symbolic dynamics

Visibility algorithms are a family of geometric and ordering criteria by which a real-valued time series of N data is mapped into a graph of N nodes. This graph has been shown to often inherit in its topology nontrivial properties of the series structure, and can thus be seen as a combinatorial representation of a dynamical system. Here we explore in some detail the relation between visibility graphs and symbolic dynamics. To do that, we consider the degree sequence of horizontal visibility graphs generated by the one-parameter logistic map, for a range of values of the parameter for which the map shows chaotic behaviour. Numerically, we observe that in the chaotic region the block entropies of these sequences systematically converge to the Lyapunov exponent of the time series. Hence, Pesin’s identity suggests that these block entropies are converging to the Kolmogorov–Sinai entropy of the physical measure, which ultimately suggests that the algorithm is implicitly and adaptively constructing phase space partitions which might have the generating property. To give analytical insight, we explore the relation k(x),x∈[0,1] that, for a given datum with value x, assigns in graph space a node with degree k. In the case of the out-degree sequence, such relation is indeed a piece-wise constant function. By making use of explicit methods and tools from symbolic dynamics we are able to analytically show that the algorithm indeed performs an effective partition of the phase space and that such partition is naturally expressed as a countable union of subintervals, where the endpoints of each subinterval are related to the fixed point structure of the iterates of the map and the subinterval enumeration is associated with particular ordering structures that we called motifs.

Analytic sets of reals and the density function in the Cantor space

We study the density function of measurable subsets of the Cantor space. Among other things, we identify a universal set $\mathcal{U}$ for $\Sigma^{1}_{1}$ subsets of $( 0 ; 1 )$ in terms of the density function; specifically $\mathcal{U}$ is the set of all pairs $( K , r )$ with $K$ compact and $r \in ( 0 ; 1 )$ being the density of some point with respect to $K$. This result yields that the set of all $K$ such that the range of its density function is $S \cup \{ 0 , 1 \}$, for some fixed uncountable analytic set $S \subseteq ( 0 ; 1 )$, is $\Pi^{1}_{2}$-complete.

Two new equivalents of Lindelöf metric spaces

AbstractIn the realm of Lindelöf metric spaces the following results are obtained in : (i) If is a Lindelöf metric space then it is both densely Lindelöf and almost Lindelöf. In addition, under the countable axiom of choice , the three notions coincide. (ii) The statement “every separable metric space is almost Lindelöf” implies that every infinite subset of has a countably infinite subset). (iii) The statement “every almost Lindelöf metric space is quasi totally bounded implies . (iv) The proposition “every quasi totally bounded metric space is separable” lies, in the deductive hierarchy of choice principles, strictly between the countable union theorem and . Likewise, the statement “every pre‐Lindelöf (or Lindelöf) metric space is separable” lies strictly between and .

Quasi-metrizability of products in ZF and equivalences of CUT(fin)

It is proved in ZF that if a collection of (quasi)-metric spaces is indexed by a countable union of finite sets, then the product of this collection is (quasi)-metrizable, while it is independent of ZF that every countable product of metrizable spaces is quasi-metrizable. It is also proved that if J is a non-empty set, while a space X, consisting of at least two points, is equipped with the co-finite topology, then the product XJ is quasi-metrizable if and only if both X and J are countable unions of finite sets. Several equivalences of CUT(fin) are deduced. It is shown that, in every model of ZF+¬CUT(fin), for an uncountable set J, the space NJ can be normal (even metrizable), a Cantor cube can be simultaneously metrizable, not second-countable and non-compact.

On Certain Dense, Uncountable Subsets of the Real Line

It is a challenge to define an uncountable set of real numbers that is dense in the real line and whose complement is also uncountable and dense. In this article we specify, for any positive integer k, a partition of the real line into k + 1 sets that are uncountable and dense; moreover, every real number is a condensation point for each of these sets. We show that one of these sets has measure one when restricted to the unit interval, while each of the others has measure zero. We then define a function that is continuous on the set of measure 1 but discontinuous elsewhere.

Uncountable equilateral sets in Banach spaces of the form C(K)

The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin’s axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form C(K) has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form C(K) which has no uncountable equilateral set (or equivalently no uncountable (1+ε)-separated set in the unit sphere for any ε > 0) making another consistent combinatorial assumption. The compact K is a version of the split interval obtained from a sequence of functions which behave in an anti-Ramsey manner. It remains open if there is an absolute example of a nonseparable Banach space of the form different than C(K) which has no uncountable equilateral set. It follows from the results of S. Mercourakis and G. Vassiliadis that our example has an equivalent renorming in which it has an uncountable equilateral set. It remains open if there are consistent examples of nonseparable Banach spaces which have no uncountable equilateral sets in any equivalent renorming but it follows from the results of S. Todorcevic that it is consistent that every nonseparable Banach space has an equivalent renorming in which it has an uncountable equilateral set.

Fixed points of Lyapunov integral operators and Gibbs measures

In this paper we shall consider the connections between Lyapunov integral operators and Gibbs measures for models with four competing interactions and uncountable (i.e. [0, 1]) set of spin values on a Cayley tree. We prove the existence of fixed points of Lyapunov integral operators and give a condition of uniqueness of a fixed point.

Nontranslation invariant Gibbs measures for models with uncountable set of spin values on a Cayley tree

We consider models with nearest-neighbour interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. It is known that the “splitting Gibbs measures” of the model can be described by solutions of a nonlinear integral equation. Recently, by solving this integral equation some periodic (in particular translation invariant) splitting Gibbs measures were found. In this paper we give three constructions of new sets of nontranslation invariant splitting Gibbs measures. Our constructions are based on known solutions of the integral equation (1.5) .

Contrast Imaging Techniques and Renal Dysfunction

Iodinated contrast has been one of the most prescribed and used drugs in contrast imaging techniques and interventional procedures However some subjects may develop contrast induced nephropathy CIN especially those in advanced chronic kidney disease CKD Diagnostic criterion is based on increasing h creatinine after receiving iodine Given the fact that there is no specific treatment for CIN prevention should be considered Uncountable actions should be taken including reducing the dose using a low osmolar substance avoiding dehydration and other nephrotoxic drugs Currently prevention has been based on using saline solution A recent paper showed that for patients with stage and of CKD sodium bicarbonate did not provided greater benefit when compared to a saline solution as well as comparing acetylcysteine to placebo Regarding to gadolinium besides nephrotoxicity is irrelevant there is a risk for developing Nephrogenic Systemic Fibrosis NSF which may occur in those with GFR lt mL min particularly appears at a GFR lt mL min Current gadolinium use guidelines are related to patients with stage GFR lt mL min according to which they should undergo hemodialysis after examination

Lipschitz Restrictions of Continuous Functions and a Simple Construction of Ulam-Zahorski C1 Interpolation

We present a simple argument showing that for every continuous function \(f\colon\mathbb{R}\to\mathbb{R}\), its restriction to some perfect set is Lipschitz. We will use this result to provide an elementary proof of the \(C^1\) free interpolation theorem, that for every continuous function \(f\colon\mathbb{R}\to\mathbb{R}\) there exists a continuously differentiable function \(g\colon\mathbb{R}\to\mathbb{R}\) which agrees with \(f\) on an uncountable set. The key novelty of our presentation is that no part of it, including the cited results, requires from the reader any prior familiarity with Lebesgue measure theory.

Approximation by entire functions on a countable union of segments on the real axis. 3. Further generalization

Approximation by entire functions on a countable union of segments on the real axis. 3. Further generalization