Discrete Subspace Research Articles

Computability of graphs

AbstractWe consider topological pairs , , which have computable type, which means that they have the following property: if X is a computable topological space and a topological imbedding such that and are semicomputable sets in X, then is a computable set in X. It is known, e.g., that has computable type if M is a compact manifold with boundary. In this paper we examine topological spaces called graphs and we show that we can in a natural way associate to each graph G a discrete subspace E so that has computable type. Furthermore, we use this result to conclude that certain noncompact semicomputable graphs in computable metric spaces are computable.

Exact smooth piecewise polynomial sequences on Alfeld splits

We develop exact piecewise polynomial sequences on Alfeld splits in any spatial dimension and any polynomial degree. An Alfeld split of a simplex is obtained by connecting the vertices of an $n$-simplex with its barycenter. We show that, on these triangulations, the kernel of the exterior derivative has enhanced smoothness. Byproducts of this theory include characterizations of discrete divergence-free subspaces for the Stokes problem, commutative projections, and simple formulas for the dimensions of smooth polynomial spaces.

The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces

Abstract We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue L p {L^{p}} -space, 1 < p < ∞ {1<p<\infty} . The greater generality of this weak setting is natural when dealing with rough data and highly irregular solutions, and when enhanced qualitative features of the approximations are needed. We first present a rigorous analysis of the well-posedness of the underlying continuous weak formulation, under natural assumptions on the advection-reaction coefficients. The main contribution is the study of several discrete subspace pairs guaranteeing the discrete stability of the method and quasi-optimality in L p {L^{p}} , and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.

Delay-Coordinate Maps and the Spectra of Koopman Operators

The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the non-mixing component of the dynamics. They factor the dynamics, which can be chaotic, into quasiperiodic rotations on tori. Here, we describe a method through which these eigenfunctions can be obtained from a kernel integral operator, which also annihilates the continuous spectrum. We show that incorporating a large number of delay coordinates in constructing the kernel of that operator results, in the limit of infinitely many delays, in the creation of a map into the discrete spectrum subspace of the Koopman operator. This enables efficient approximation of Koopman eigenfunctions from high-dimensional data in systems with pure point or mixed spectra.

A note on discrete C-embedded subspaces

Abstract It is shown that in some non-discrete topological spaces, discrete subspaces with certain cardinality are C-embedded. In particular, this generalizes the well-known fact that every countable subset of P-spaces are C-embedded. In the presence of the measurable cardinals, we observe that if X is a discrete space then every subspace of υ X (i.e., the Hewitt realcompactification of X) whose cardinal is nonmeasurable, is a C-embedded, discrete realcompact subspace of υ X. This generalizes the well-known fact that the discrete spaces with nonmeasurable cardinal are realcompact.

Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization

Abstract We consider Hausdorff discretization from a metric space E to a discrete subspace D, which associates to a closed subset F of E any subset S of D minimizing the Hausdorff distance between F and S; this minimum distance, called the Hausdorff radius of F and written rH (F), is bounded by the resolution of D. We call a closed set F separated if it can be partitioned into two non-empty closed subsets F 1 and F 2 whose mutual distances have a strictly positive lower bound. Assuming some minimal topological properties of E and D (satisfied in ℝ n and ℤ n ), we show that given a non-separated closed subset F of E, for any r > rH (F), every Hausdorff discretization of F is connected for the graph with edges linking pairs of points of D at distance at most 2r. When F is connected, this holds for r = rH (F), and its greatest Hausdorff discretization belongs to the partial connection generated by the traces on D of the balls of radius rH (F). However, when the closed set F is separated, the Hausdorff discretizations are disconnected whenever the resolution of D is small enough. In the particular case where E = ℝ n and D = ℤ n with norm-based distances, we generalize our previous results for n = 2. For a norm invariant under changes of signs of coordinates, the greatest Hausdorff discretization of a connected closed set is axially connected. For the so-called coordinate-homogeneous norms, which include the Lp norms, we give an adjacency graph for which all Hausdorff discretizations of a connected closed set are connected.

Subspace mixed rational time-frequency multiwindow Gabor frames and their Gabor duals

For a usual multiwindow Gabor system, all windows share common time-frequency shifts. A mixed multiwindow Gabor system is one of its generalizations, for which time-frequency shifts vary with the windows. This paper addresses subspace mixed multiwindow Gabor systems with rational time-frequency product lattices. It is a continuation of (Li and Zhang in Abstr. Appl. Anal. 2013:357242, 2013; Zhang and Li in J. Korean Math. Soc. 51:897–918, 2014). In (Li and Zhang in Abstr. Appl. Anal. 2013:357242, 2013) we dealt with discrete subspace mixed Gabor systems and in (Zhang and Li in J. Korean Math. Soc. 51:897–918, 2014) with L^{2}(mathbb{R}) ones. In this paper, using a suitable Zak transform matrix method, we characterize subspace mixed multiwindow Gabor frames and their Gabor duals, obtain explicit expressions of Gabor duals, and characterize the uniqueness of Gabor duals. We also provide some examples, which show that there exist significant differences between mixed multiwindow Gabor frames and usual multiwindow Gabor frames.

If K is Gul'ko compact, then every iterated function space C,(K) has a uniformly dense subspace of countable pseudocharacter

We establish that C p ( X ) has a dense subspace of countable i -weight if and only if d ( C p ( X ) ) ⩽ c . It is also proved that the space C p ( K ) has a dense F σ -discrete subspace whenever K is Corson compact. If both spaces X and C p ( X ) are Lindelöf Σ, then for any natural n ⩾ 2 , the iterated function space C p , n ( X ) has a uniformly dense subspace of countable pseudocharacter. In the case of a Gul'ko compact space K , there is a uniformly dense subspace of countable pseudocharacter in C p , n ( K ) for any n ∈ N . This is a new result even for C p ( K ) given that K is Eberlein compact.

Remarks on ω-domination of discrete subspaces

Given a space X, we will say that a class of subsets of X is dominated by a class Ɓ if for any A ∈ , there exists a B ∈ Ɓ such that A ⊂ . In particular, all (closed) discrete subsets of X are countably dominated (which we frequently abbreviate as ω-dominated) if, for any (closed) discrete set D ⊂ X, there exists a countable set B ⊂ X such that D ⊂ . In this paper, we investigate the topological properties of spaces in which (closed) discrete subspaces are dominated either by countable subsets or by Lindelöf subspaces.

Star countable spaces and $$\omega $$ ω -domination of discrete subspaces

We establish that a first countable $$\omega $$ -monolithic space is star countable if and only if it has countable extent. A consistent example is given of a first countable normal star Lindelof space of uncountable extent. Under the continuum hypothesis we prove that for any compact K, the space $$C_p(K)$$ is star countable if and only if it is Lindelof. The above-mentioned results answer several published open questions.

Weakly linearly lindelöf monotonically normal spaces are lindelöf

A space X is weakly linearly Lindelöf if for any family U of non-empty open subsets of X of regular uncountable cardinality κ, there exists a point x ∈ X such that every neighborhood of x meets κ-many elements of U. We also introduce the concept of almost discretely Lindelöf spaces as the ones in which every discrete subspace can be covered by a Lindelöf subspace. We prove that, in addition to linearly Lindelöf spaces, both weakly Lindelöf spaces and almost discretely Lindelöf spaces are weakly linearly Lindelöf. The main result of the paper is formulated in the title. It implies that every weakly Lindelöf monotonically normal space is Lindelöf, a result obtained earlier in [3]. We show that, under the hypothesis 2ω < ωω, if the co-diagonal ΔcX = (X × X) \ΔX is discretely Lindelöf, then X is Lindelöf and has a weaker second countable topology; here ΔX = {(x, x): x ∈ X} is the diagonal of the space X. Moreover, discrete Lindelöfness of ΔcX together with the Lindelöf Σ-property of X imply that X has a countable network.

The Samuel realcompactification of a metric space

In this paper we introduce a realcompactification for any metric space (X,d), defined by means of the family of all its real-valued uniformly continuous functions. We call it the Samuel realcompactification, according to the well known Samuel compactification associated to the family of all the bounded real-valued uniformly continuous functions. Among many other things, we study the corresponding problem of the Samuel realcompactness for metric spaces. At this respect, we prove that a result of Katětov–Shirota type occurs in this context, where the completeness property is replaced by Bourbaki-completeness (a notion recently introduced by the authors) and the closed discrete subspaces are replaced by the uniformly discrete ones. More precisely, we see that a metric space (X,d) is Samuel realcompact iff it is Bourbaki-complete and every uniformly discrete subspace of X has non-measurable cardinal. As a consequence, we derive that a normed space is Samuel realcompact iff it has finite dimension. And this means in particular that realcompactness and Samuel realcompactness can be very far apart. The paper also contains results relating this realcompactification with the so-called Lipschitz realcompactification (also studied here), with the classical Hewitt–Nachbin realcompactification and with the completion of the initial metric space.

Continuous extension of functions from countable sets

We give a characterization of countable discrete subspace A of a topological space X such that there exists a (linear) continuous mapping φ:Cp⁎(A)→Cp(X) with φ(y)|A=y for every y∈Cp⁎(A). Using this characterization we answer two questions of A. Arhangel'skii. Moreover, we introduce the notion of well-covered subset of a topological space and prove that for well-covered functionally closed subset A of a topological space X there exists a linear continuous mapping φ:Cp(A)→Cp(X) with φ(y)|A=y for every y∈Cp(A).

Rosenthal compacta that are premetric of finite degree

Despite its analytical origins, Rosenthal compacta have been extensively studied from other areas like general topology and its study involves techniques from Logic, Ramsey Theory and Descriptive Set Theory. In this regard we can highlight a celebrated result on the structure of the class of separable Rosenthal compacta due to S. Todorcevic [Tod99]. Namely, Todorcevic proved that a separable Rosenthal compactum either contains a discrete subspace of size continuum or it is a pre-metric compactum of degree at most two. Moreover, the author proves that such kind of compacta contain either copies of the Split Interval S(I) or copies of the Alexadroff Duplicate of the Cantor space D(2N).

Some sufficiency conditions for D-spaces, dually discrete and its applications

In this note, we obtain some sufficiency conditions for D-spaces, dually discrete, and dually scattered of rank≤2. As an application, we show that if a space X has a chain (F) point network W={W(x):x∈X} such that W(x) is a chain and is a closure-preserving family of subsets of X for each x∈X then X is a paracompact D-space. If a space X has an ω1-sheltering chain (F) point network W={W(x):x∈X} such that W(x) is a chain and is a weak ω-closure-preserving family of subsets of X for each x∈X, then X is a paracompact D-space.If X is a chain neighborhood (F) space, then for each neighborhood assignment ϕ for X there is an open neighborhood Vx of x for each x∈X such that x∈Vx⊂ϕ(x) and for each closed discrete subspace F1 of X and for each closed subset A of X, there exists a closed discrete subspace D of X such that D⊂A∩{z∈X∖ϕ(F1):Vz∩F1≠∅} and A∩{z∈X∖ϕ(F1):Vz∩F1≠∅}⊂ϕ(D). By the above conclusion and a sufficiency condition for D-spaces in this note, we can get a known conclusion that every chain neighborhood (F) space is a D-space [8].

A note on the Menger (Rothberger) property and topological games

In this note we show that a T1 topological space X is a Menger space if and only if for each sequence {ϕn:n∈ω} of neighborhood assignments for X, there exists, for each n∈ω, a finite subset Dn of X such that X=⋃{ϕn(d):d∈Dn,n∈ω} and D=⋃{Dn:n∈ω} is a closed discrete subspace of X. A T1 topological space X is a Rothberger space if and only if for each sequence {ϕn:n∈ω} of neighborhood assignments for X, there exists, for each n∈ω, a point dn∈X such that X=⋃{ϕn(dn):n∈ω} and D={dn:n∈ω} is a closed discrete subspace of X.Let M be the class of all spaces with the Menger property. Let R be the class of all Rothberger spaces. We show that a M-like space has the Menger property. A R-like space is a Rothberger space.Let C be the class of all compact spaces. Let W be the class of all countable spaces. We prove that a finite product of C-like spaces is a C-like space. As a corollary we know that a finite product of C-like spaces is a Menger space. In the last part of this note we show that a finite product of W-like Hausdorff spaces is a nc-W-like space. A finite product of W-like spaces is a Rothberger space.

Interatomic Coulombic decay widths of helium trimer: Ab initio calculations.

We report on an extensive study of interatomic Coulombic decay (ICD) widths in helium trimer computed using a fully ab initio method based on the Fano theory of resonances. Algebraic diagrammatic construction for one-particle Green's function is utilized for the solution of the many-electron problem. An advanced and universal approach to partitioning of the configuration space into discrete states and continuum subspaces is described and employed. Total decay widths are presented for all ICD-active states of the trimer characterized by one-site ionization and additional excitation of an electron into the second shell. Selected partial decay widths are analyzed in detail, showing how three-body effects can qualitatively change the character of certain relaxation transitions. Previously unreported type of three-electron decay processes is identified in one class of the metastable states.

Equilibration a Posteriori Error Estimation for Convection–Diffusion–Reaction Problems

We study a posteriori error estimates for convection---diffusion---reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation, we derive reliable and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of $${H({{\mathrm{div}}},\Omega )}$$H(div,Ω). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of the convection part of the differential operator, robustness of the error estimator with respect to the coefficients of the problem is achieved. Numerical benchmarks illustrate the good performance of the error estimators for singularly perturbed problems, in particular with dominating convection.

A note on joint metrizability of spaces on families of subspaces

In this note, we introduce concepts of JSM-spaces and JADM-spaces following a general idea of Arhangel'skii and Shumrani. Let X be a topological space. Denote FS(X)={S∪{xS}: S is a convergent sequence of X which converges to the point xS}. A subspace Y of X is called almost discrete if Y has at most one non-isolated point. Denote FAD(X)={Y:Y is an almost discrete subspace of X}. A space X is called a JSM-space (JADM-space) if there is a metric d on the set X such that d metrizes all subspaces of X which belong to FS(X) (FAD(X)). We get some conclusions on JSM-spaces and JADM-spaces.A space X is said to be weak Fréchet if for any σ-closed discrete subspace B of X with x∈B¯ there is a sequence {xn:n∈N}⊂B which converges to x. We show that if X is a Hausdorff weak Fréchet compactly metrizable strong Σ⁎-space then X has a σ-locally finite network.

Discrete reflexivity in function spaces

We study systematically when $C_p(X)$ has a topological property $\mathcal{P}$ if $C_p(X)$ is discretely $\mathcal{P}$, i.e., the set $\overline {D}$ has $\mathcal{P}$ for every discrete subspace $D\subset C_p(X)$. We prove that it is independent of ZFC whether discrete metrizability of $C_p(X)$ implies its metrizability for a compact space $X$. We show that it is consistent with ZFC that countable tightness and Lindelöf $\Sigma$-property are not discretely reflexive in spaces $C_p(X)$. It is also established that a space $X$ must be countable and discrete if $C_p(X)$ is discretely Čech-complete. If $C_p(X)$ is discretely $\sigma$-compact then $X$ has to be finite.