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

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}}-space,{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}}, and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.

Embeddings in the Fell and Wijsman topologies

It is shown that if a T2 topological space X contains a closed uncountable discrete subspace, then the spaces (?1 + 1)? and (?1 + 1)?1 embed into (CL(X),?F), the hyperspace of nonempty closed subsets of X equipped with the Fell topology. If (X, d) is a non-separable perfect topological space, then (?1 + 1)? and (?1 +1)?1 embed into (CL(X), ?w(d)), the hyperspace of nonempty closed subsets of X equipped with the Wijsman topology, giving a partial answer to the Question 3.4 in [2].

Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization

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 F1 and F2 whose mutual distances have a strictly positive lower bound. Assuming some minimal topological properties of E and D (satisfied in ℝn and ℤn), 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 = ℝn and D = ℤn 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.

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.

Discrete Subspace Multiwindow Gabor Frames and Their Duals

This paper addresses discrete subspace multiwindow Gabor analysis. Such a scenario can model many practical signals and has potential applications in signal processing. In this paper, using a suitable Zak transform matrix we characterize discrete subspace mixed multi-window Gabor frames (Riesz bases and orthonormal bases) and their duals with Gabor structure. From this characterization, we can easily obtain frames by designing Zak transform matrices. In particular, for usual multi-window Gabor frames (i.e., all windows have the same time-frequency shifts), we characterize the uniqueness of Gabor dual of type I (type II) and also give a class of examples of Gabor frames and an explicit expression of their Gabor duals of type I (type II).

Topology for Computations of Concurrent Automata

We generalize the results on α-complex traces from [14, 16] to the realm of concurrent automata. We show that the α-complex computations can be defined as elements of a compact, totally disconnected and complete topological semigroup containing the finite computations as a dense and discrete subspace. But in this semigroup it is not possible to define infinite products as limits of finite ones in a satisfactory way. Therefore, we define a second semigroup of restricted α-complex computations. The underlying set of this semigroup carries a metric such that it becomes compact and complete. Furthermore, the finite computations are a dense and discrete subset and the multiplication is uniformly continuous not on the whole semigroup but on the set of finite computations. Finally, here we can define infinite products as limits of finite ones.

Reflexive objects in topological categories

This paper presents several basic results about the non-existence of reflexive objects in cartesian closed topological categories of Hausdorff spaces. In particular, we prove that there are no non-trivial countably compact reflexive objects in the category of Hausdorff k-spaces and, more generally, that any non-trivial reflexive Tychonoff space in this category contains a closed discrete subspace corresponding to a numeral system in the sense of Wadsworth. In addition, we establish that a reflexive Tychonoff space in a cartesian-closed topological category cannot contain a non-trivial continuous image of the unit interval. Therefore, if there exists a non-trivial reflexive Tychonoff space, it does not have a nice geometric structure.

Subsemigroups of Stone-Čech compactifications

The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large topological space; for example, any countable discrete subspace of the growth ℕ* = βℕ/ℕ has a closure which is homeomorphic to βℕ itself ([23], §3·5] Now ℕ, while hardly inspiring as a discrete topological space, has a rich algebrai structure. That βℕ also has a semigroup structure which extends that of (ℕ, +) and in which multiplication is continuous in one variable has been apparent for about 30 years. (Civin and Yood [3] showed that βG was a semigroup for each discrete group G, and any mathematician could then have spotted that βℕ was a subsemigroup of βℕ.) The question which now appears natural was explicitly raised by van Douwen[6] in 1978 (in spite of the recent publication date of his paper), namely, does ℕ* contain subspaces simultaneously algebraically isomorphic and homeomorphic to βℕ? Progress on this question was slight until Strauss [22] solved it in a spectacular fashion: the image of any continuous homomorphism from βℕ into ℕ* must be finite, and so the homomorphism cannot be injective. This dramatic advance is not the end of the story. It is still not known whether that image can contain more than one point. Indeed, what appears to be one of the most difficult questions about the algebraic structure of βℕ is whether it contains any non-trivial finite subgroups

Canonization theorems and applications

We improve the canonization theorems generalizing the Erdös-Rado theorem, and as a result complete the answer to “When does a Hausdorff space of cardinality χ necessarily have a discrete subspace of cardinality k” We also improve the results on existence of free subsets.

On hereditarily Lindelöf spaces

This paper considers the question of when a space with the property that each discrete subspace is countable is hereditarily Lindelbölf. The question is answered affirmatively for the class of ROP spaces and for the class of hereditarily meta-Lindelöf spaces. A characterization of hereditarily Lindelöf spaces in terms of countable subspaces is given.