Zero-dimensional Research Articles

Free Subspaces of Free Locally Convex Spaces

IfXandYare Tychonoff spaces, letL(X)andL(Y)be the free locally convex space overXandY, respectively. For generalXandY, the question of whetherL(X)can be embedded as a topological vector subspace ofL(Y)is difficult. The best results in the literature are that ifL(X)can be embedded as a topological vector subspace ofL(I), whereI=[0,1], thenXis a countable-dimensional compact metrizable space. Further, ifXis a finite-dimensional compact metrizable space, thenL(X)can be embedded as a topological vector subspace ofL(I). In this paper, it is proved thatL(X)can be embedded inL(R)as a topological vector subspace ifXis a disjoint union of a countable number of finite-dimensional locally compact separable metrizable spaces. This is the case ifX=Rn, n∈N.It is also shown that ifGandQdenote the Cantor space and the Hilbert cubeIN, respectively, then (i)L(X)is embedded inL(G)if and only ifXis a zero-dimensional metrizable compact space; (ii)L(X)is embedded inL(Q)if and only ifYis a metrizable compact space.

Zero-dimensional transient model of large-scale cooling ponds using well-mixed approach

Nowadays, nuclear power plants around the world produce vast amounts of spent fuel. After discharge, it requires adequate cooling to prevent radioactive materials being released into the environment. One of the systems available to provide such cooling is the spent fuel cooling pond. The recent incident at Fukushima, Japan shows that these cooling ponds are associated with safety concerns and scientific studies are required to analyse their thermal performance. However, the modelling of spent fuel cooling ponds can be very challenging. Due to their large size and the complex phenomena of heat and mass transfer involved in such systems. In the present study, we have developed a zero-dimensional (Z-D) model based on the well-mixed approach for a large-scale cooling pond. This model requires low computational time compared with other methods such as computational fluid dynamics (CFD) but gives reasonable results are key performance data. This Z-D model takes into account the heat transfer processes taking place within the water body and the volume of humid air above its surface as well as the ventilation system. The methodology of the Z-D model was validated against data collected from existing cooling ponds. A number of studies are conducted considering normal operating conditions as well as in a loss of cooling scenario. Moreover, a discussion of the implications of the assumption to neglect heat loss from the water surface in the context of large-scale ponds is also presented. Also, a sensitivity study is performed to examine the effect of weather conditions on pond performance.

Dynamics in dimension zero A survey

The goal of this paper is to put together several techniques in handling dynamical systems on zero-dimensional spaces, such as array representation, inverse limit representation, or Bratteli-Vershik representation. We describe how one can switch from one representation to another. We also briefly review some more recent related notions: symbolic extensions, symbolic extensions with an embedding, and uniform generators. We devote a great deal of attention to marker techniques and we use them to prove two types of results: one concerning entropy and vertical data compression, and another, about the existence of isomorphic minimal models for aperiodic systems. We also introduce so-called decisiveness of Bratteli-Vershik systems and give for it a sufficient condition.

On disconnected images of connected spaces

We introduce the notion that a zero-dimensional compact space L is a Boolean image of an arbitrary compact space K. When K is also zero-dimensional, this just means that L is a continuous image of K. However, a number of interesting questions arise when we consider connected compacta K.

Minimal models for actions of amenable groups

We prove that on a metrizable, compact, zero-dimensional space every free action of an amenable group is measurably isomorphic to a minimal G -action with the same, i.e. affinely homeomorphic, simplex of measures.

Some remarks on Baire’s grand theorem

We provide a game theoretical proof of the fact that if f is a function from a zero-dimensional Polish space to $$ \mathbb N^{\mathbb N}$$ that has a point of continuity when restricted to any non-empty compact subset, then f is of Baire class 1. We use this property of the restrictions to compact sets to give a generalisation of Baire’s grand theorem for functions of any Baire class.

Solving piecewise polynomial constraint systems with decomposition and a subdivision-based solver

Piecewise polynomial constraint systems are common in numerous problems in computational geometry, such as constraint programming, modeling, and kinematics. We propose a framework that is capable of decomposing, and efficiently solving a wide variety of complex piecewise polynomial constraint systems, that include both zero constraints and inequality constraints, with zero-dimensional or univariate solution spaces. Our framework combines a subdivision-based polynomial solver with a decomposition algorithm in order to handle large and complex systems. We demonstrate the capabilities of our framework on several types of problems and show its performance improvement over a state-of-the-art solver.

Topological properties preserved by weakly discontinuous maps and weak homeomorphisms

A map f:X→Y between topological spaces is called weakly discontinuous if each subspace A⊂X contains an open dense subset U⊂A such that the restriction f|U is continuous. A bijective map f:X→Y between topological spaces is called a weak homeomorphism if f and f−1 are weakly discontinuous. We study properties of topological spaces preserved by weakly discontinuous maps and weak homeomorphisms. In particular, we show that weak homeomorphisms preserve network weight, hereditary Lindelöf number, dimension. Also we classify infinite zero-dimensional σ-Polish metrizable spaces up to a weak homeomorphism and prove that any such space X is weakly homeomorphic to one of 9 spaces: ω, 2ω, Nω, Q, Q⊕2ω, Q×2ω, Q⊕Nω, (Q×2ω)⊕Nω, Q×Nω.

Classifying homogeneous cellular ordinal balleans up to coarse equivalence

For every ballean $X$ we introduce two cardinal characteristics $cov^\flat(X)$ and $cov^\sharp(X)$ describing the capacity of balls in $X$. We observe that these cardinal characteristics are invariant under coarse equivalence and prove that two cellular ordinal balleans $X,Y$ are coarsely equivalent if $cof(X)=cof(Y)$ and $cov^\flat(X)=cov^\sharp(X)=cov^\flat(Y)=cov^\sharp(Y)$. This result implies that a cellular ordinal ballean $X$ is homogeneous if and only if $cov^\flat(X)=cov^\sharp(X)$. Moreover, two homogeneous cellular ordinal balleans $X,Y$ are coarsely equivalent if and only if $cof(X)=cof(Y)$ and $cov^\sharp(X)=cov^\sharp(Y)$ if and only if each of these balleans coarsely embeds into the other ballean. This means that the coarse structure of a homogeneous cellular ordinal ballean $X$ is fully determined by the values of the cardinals $cof(X)$ and $cov^\sharp(X)$. For every limit ordinal $\gamma$ we shall define a ballean $2^{<\gamma}$ (called the Cantor macro-cube), which in the class of cellular ordinal balleans of cofinality $cf(\gamma)$ plays a role analogous to the role of the Cantor cube $2^{\kappa}$ in the class of zero-dimensional compact Hausdorff spaces. We shall also present a characterization of balleans which are coarsely equivalent to $2^{<\gamma}$. This characterization can be considered as an asymptotic analogue of Brouwer's characterization of the Cantor cube $2^\omega$.

Baire classification of fragmented maps and approximation of separately continuous functions

We study a class of fragmented maps which contains all barely continuous maps, scatteredly continuous maps and maps which are pointwise discontinuous on each closed set. We prove that for any Hausdorff space X, a metric space Z and a fragmented map \(f:X\rightarrow Z\) there exists a pointwisely convergent to f sequence of continuous functions \(f_n:X\rightarrow Z\) with an additional condition (which is a weakening of the uniform convergence at each point) in the following cases: (a) X is stratifiable and Z is locally convex equiconnected; (b) X is stratifiable with \(\dim X<\infty \) and Z is equiconnected; (c) X is stratifiable with \(\dim X=0\). We show that for a hereditarily Baire space X, a compact space Y and a metric space Z every separately continuous map Open image in new window is a pointwise limit of a sequence of continuous maps which is uniformly convergent to f with respect to each variable at every point of Open image in new window in cases (a)–(c); (d) X is a perfect zero-dimensional compact space; and (e) X is the Sorgenfrey line.

Vector-valued modular forms and the mock theta conjectures

The mock theta conjectures are ten identities involving Ramanujan’s fifth-order mock theta functions. The conjectures were proven by Hickerson in 1988 using q-series methods. Using methods from the theory of harmonic Maass forms, specifically work of Zwegers and Bringmann–Ono, Folsom reduced the proof of the mock theta conjectures to a finite computation. Both of these approaches involve proving the identities individually, relying on work of Andrews–Garvan. Here we give a unified proof of the mock theta conjectures by realizing them as an equality between two nonholomorphic vector-valued modular forms which transform according to the Weil representation. We then show that the difference of these vectors lies in a zero-dimensional vector space.

Functionally countable subalgebras and some properties of the Banaschewski compactification

Let $X$ be a zero-dimensional space and $C_c(X)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K(X)$ (resp., $C_c^{\psi}(X))$ the set of all functions in $C_c(X)$ with compact (resp., pseudocompact) support. First, we observe that $C_c^K(X)=O_c^{\beta_0X\setminus X}$ (resp., $C^{\psi}_c(X)=M_c^{\beta_0X\setminus \upsilon_0X}$), where $\beta_0X$ is the Banaschewski compactification of $X$ and $\upsilon_0X$ is the $\mathbb{N}$-compactification of $X$. This implies that for an $\mathbb{N}$-compact space $X$, the intersection of all free maximal ideals in $C_c(X)$ is equal to $C_c^K(X)$, i.e., $M_c^{\beta_0X\setminus X}=C_c^K(X)$. By applying methods of functionally countable subalgebras, we then obtain some results in the remainder of the Banaschewski compactification. We show that for a non-pseudocompact zero-dimensional space $X$, the set $\beta_0X\setminus \upsilon_0X$ has cardinality at least $2^{2^{\aleph_0}}$. Moreover, for a locally compact and $\mathbb{N}$-compact space $X$, the remainder $\beta_0X\setminus X$ is an almost $P$-space. These results lead us to find a class of Parovicenko spaces in the Banaschewski compactification of a non pseudocompact zero-dimensional space. We conclude with a theorem which gives a lower bound for the cellularity of the subspaces $\beta_0X\setminus \upsilon_0X$ and $\beta_0X\setminus X$, whenever $X$ is a zero-dimensional, locally compact space which is not pseudocompact.

Estimability and dependency analysis of model parameters based on delay coordinates.

In data-driven system identification, values of parameters and not observed variables of a given model of a dynamical system are estimated from measured time series. We address the question of estimability and redundancy of parameters and variables, that is, whether unique results can be expected for the estimates or whether, for example, different combinations of parameter values would provide the same measured output. This question is answered by analyzing the null space of the linearized delay coordinates map. Examples with zero-dimensional, one-dimensional, and two-dimensional null spaces are presented employing the Hindmarsh-Rose model, the Colpitts oscillator, and the Rössler system.

Measures and fibers

We study measures on compact spaces by analyzing the properties of fibers of continuous mappings into 2ω. We show that if a compact zerodimensional space K carries a measure of uncountable Maharam type, then such a mapping has a non-scattered fiber and, if we assume additionally a weak version of Martin's Axiom, such a mapping has a fiber carrying a measure of uncountable Maharam type. Also, we prove that every compact zerodimensional space which supports a strictly positive measure and which can be mapped into 2ω by a finite-to-one function is separable.

Inverse limits with bonding functions whose graphs are connected

First, answering a question by Roškarič and Tratnik, we present inverse sequences of simple triods or simple closed curves with set-valued bonding functions whose graphs are arcs and the limits are n-point sets. Second, we present a wide class of zero-dimensional spaces that can be obtained as the inverse limits of arcs with one set-valued function whose graph is an arc.

On epireflective and coreflective hulls of a particular L-topological space

In this paper, we study some aspects of the category L-ZTop of zero-dimensional L-topological spaces. After noting that it is a topological category, we identify a ‘Sierpinski object’ LZ in it. We further show that two epireflective hulls of LZ respectively turn out to be the categories of zero-dimensional T0-L-topological spaces and of zero-dimensional sober L-topological spaces. We also determine the coreflective hull of LZ in the category of L-topological spaces.

Tilings of amenable groups

Abstract We prove that for any infinite countable amenable group G, any ε &gt; 0 {\varepsilon&gt;0} and any finite subset K ⊂ G {K\subset G} , there exists a tiling (partition of G into finite “tiles” using only finitely many “shapes”), where all the tiles are ( K , ε ) {(K,\varepsilon)} -invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of G (in the sense that the mappings, associated to elements of G other than the unit, have no fixed points) on a zero-dimensional space, such that the topological entropy of this action is zero.

Remarks on $LBI$-subalgebras of $C(X)$

Let $A(X)$ denote a subalgebra of $C(X)$ which is closed under local bounded inversion, briefly, an $LBI$-subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151--163. By characterizing maximal ideals of $A(X)$, we generalize the notion of $z_A^\beta$-ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C(X)$ containing $C^*(X)$, Comment. Math. Univ. Carolin. 48 (2007), 273--280 for intermediate subalgebras, to the $LBI$-subalgebras. Using these, it is simply shown that the structure space of every $LBI$-subalgebra is homeomorphic with a quotient of $\beta X$. This gives a different approach to the results of Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151--163 and also shows that the Banaschewski-compactification of a zero-dimensional space $X$ is a quotient of $\beta X$. Finally, we consider the class of complete rings of functions which was first defined in Byun H.L., Redlin L., Watson S., Local invertibility in subrings of $C^*(X)$, Bull. Austral. Math. Soc. 46 (1992), 449--458. Showing that every such subring is an $LBI$-subalgebra, we prove that the compactification of $X$ associated to each complete ring of functions, which is identified in Byun H.L., Redlin L., Watson S., Local invertibility in subrings of $C^*(X)$, Bull. Austral. Math. Soc. 46 (1992), 449--458 via the mapping ${\mathcal Z}_A$, is in fact, the structure space of that subring. Henceforth, some statements in Byun H.L., Redlin L., Watson S., Local invertibility in subrings of $C^*(X)$, Bull. Austral. Math. Soc. 46 (1992), 449--458 could be proved in a different way.

On a question of $C_c(X)$

In this short article we answer the question posed in Ghadermazi M., Karamzadeh O.A.S., Namdari M., On the functionally countable subalgebra of $C(X)$, Rend. Sem. Mat. Univ. Padova 129 (2013), 47--69. It is shown that $C_c(X)$ is isomorphic to some ring of continuous functions if and only if $\upsilon_0 X$ is functionally countable. For a strongly zero-dimensional space $X$, this is equivalent to say that $X$ is functionally countable. Hence for every $P$-space it is equivalent to pseudo-$\aleph_0$-compactness.

Topological properties of function spaces Ck(X,2) over zero-dimensional metric spaces X

Let X be a zero-dimensional metric space and X′ its derived set. We prove the following assertions: (1) the space Ck(X,2) is an Ascoli space iff Ck(X,2) is kR-space iff either X is locally compact or X is not locally compact but X′ is compact, (2) Ck(X,2) is a k-space iff either X is a topological sum of a Polish locally compact space and a discrete space or X is not locally compact but X′ is compact, (3) Ck(X,2) is a sequential space iff X is a Polish space and either X is locally compact or X is not locally compact but X′ is compact, (4) Ck(X,2) is a Fréchet–Urysohn space iff Ck(X,2) is a Polish space iff X is a Polish locally compact space, (5) the space Ck(X,2) is normal iff X′ is separable, (6) Ck(X,2) has countable tightness iff X is separable. In cases (1)–(3) we obtain also a topological and algebraic structure of Ck(X,2).