Dense Subspace Research Articles

COUNTABLE SPACES, REALCOMPACTNESS, AND THE PSEUDOINTERSECTION NUMBER

Abstract All spaces are assumed to be Tychonoff. Given a realcompact space X, we denote by $\mathsf {Exp}(X)$ the smallest infinite cardinal $\kappa $ such that X is homeomorphic to a closed subspace of $\mathbb {R}^\kappa $ . Our main result shows that, given a cardinal $\kappa $ , the following conditions are equivalent: • There exists a countable crowded space X such that $\mathsf {Exp}(X)=\kappa $ . • $\mathfrak {p}\leq \kappa \leq \mathfrak {c}$ . In fact, in the case $\mathfrak {d}\leq \kappa \leq \mathfrak {c}$ , every countable dense subspace of $2^\kappa $ provides such an example. This will follow from our analysis of the pseudocharacter of countable subsets of products of first-countable spaces. Finally, we show that a scattered space of weight $\kappa $ has pseudocharacter at most $\kappa $ in any compactification. This will allow us to calculate $\mathsf {Exp}(X)$ for an arbitrary (that is, not necessarily crowded) countable space X.

On a variant of extremal disconnectedness in locales and spaces

We study a variant of extremal disconnectedness in frames, defined by requiring that any two disjoint open sublocales should be separable by disjoint open sublocales each of which is the interior of some zero-sublocale. In spaces, this notion was recently introduced in [1] in order to study some properties of frames of z-ideals of rings of continuous functions. The point-free approach adopted here brings to the fore some topological results not considered in [1], such as that a dense z-embedded subspace of a Tychonoff space has this property if and only if the space containing it has the property. In [1], this was proved only for the embedding X ⊆ βX. We give a ring-theoretic characterization by defining a property of f-rings in a transparent manner which shows why the property should be defined the way we do. The property is that if two annihilator ideals meet trivially, then the annihilator of each of the annihilator ideals contains a positive element such that the sum of the two elements is a non-zerodivisor.

On countably saturated and pre-dyadic spaces

In this paper, we introduce a new class of spaces which extends the class of dyadic compacta. This permits to extend a classical theorem of Engelking and Pełczyński. Technically, our arguments are based on the concept of a countably saturated space. This is a space in which the closure of every countable subset is a metrizable compactum. The existence of a dense countably saturated subspace in compact Hausdorff spaces is shown below to play a crucial role.

Multimorphisms on pre-Riesz spaces of continuous functions

In the theory of vector lattices, mostly three classes of lattice ordered algebras are investigated, called f-algebras, d-algebras, and almost-f-algebras. Each class is endowed with a different type of positive bilinear map as the multiplication, namely with bi-orthomorphisms, bi-Riesz homomorphisms, and orthosymmetric maps, respectively. On the space C(K), with K a compact Hausdorff space, representations as weighted composition or integral operators are known. We give generalized definitions of the above mentioned maps in the setting of partially ordered vector spaces, where we deal with n-linear maps. We generalize the representations to the case of certain subspaces of some C(K) space, with K a compact Hausdorff space. With the representations at hand, we deduce basic properties of these maps. The results also cover the case of order unit spaces, which can be seen as order dense subspaces of some C(K) space via the functional representation.

Extended state space for describing renormalized Fock spaces in QFT

We revisit the non-perturbative renormalization of a class of simple polaron models with resting fermions. The considered dispersion relations and form factors are allowed to be highly singular, such that infinite self-energies and wave function renormalizations may occur and the diagonalizing dressing transformations might not be implementable on Fock space. Instead of taking cutoffs, we rigorously interpret the self-energies and wave function renormalizations as elements of suitable vector spaces, as well as a field extension of the complex numbers. Moreover, we define two extended state vector spaces (ESSs) over this field extension, which contain a dense subspace of Fock space, but also incorporate non-square-integrable wave functions. Elements of these ESSs can be seen as states of virtual particles, described in a mathematically rigorous way. The Hamiltonian without cutoffs, formally infinite counterterms, and the dressing transformation can then be defined as linear operators between certain subspaces of the two Fock space extensions. This way, we obtain a renormalized Hamiltonian which can be realized as a densely defined self-adjoint operator on Fock space.

Bridging Dense and Sparse Maximum Inner Product Search

Maximum inner product search (MIPS) over dense and sparse vectors have progressed independently in a bifurcated literature for decades; the latter is better known as top- \(k\) retrieval in Information Retrieval. This duality exists because sparse and dense vectors serve different end goals. That is despite the fact that they are manifestations of the same mathematical problem. In this work, we ask if algorithms for dense vectors could be applied effectively to sparse vectors, particularly those that violate the assumptions underlying top- \(k\) retrieval methods. We study clustering-based approximate MIPS where vectors are partitioned into clusters and only a fraction of clusters are searched during retrieval. We conduct a comprehensive analysis of dimensionality reduction for sparse vectors, and examine standard and spherical KMeans for partitioning. Our experiments demonstrate that clustering-based retrieval serves as an efficient solution for sparse MIPS. As byproducts, we identify two research opportunities and explore their potential. First, we cast the clustering-based paradigm as dynamic pruning and turn that insight into a novel organization of the inverted index for approximate MIPS over general sparse vectors. Second, we offer a unified regime for MIPS over vectors that have dense and sparse subspaces, that is robust to query distributions.

Unbounded Order Convergence in Ordered Vector Spaces

We consider an ordered vector space X. We define the net xα⊆X to be unbounded order convergent to x (denoted as xα⟶uox). This means that for every 0≤y∈X, there exists a net yβ (potentially over a different index set) such that yβ↓0, and for every β, there exists α0 such that ±xα−xu,yl⊆yβl whenever α≥α0. The emergence of a broader convergence, stemming from the recognition of more ordered vector spaces compared to lattice vector spaces, has prompted an expansion and broadening of discussions surrounding lattices to encompass additional spaces. We delve into studying the properties of this convergence and explore its relationships with other established order convergence. In every ordered vector space, we demonstrate that under certain conditions, every uo-convergent net implies uo-Cauchy, and vice versa. Let X be an order dense subspace of the directed ordered vector space Y. If J⊆Y is a uo-band in Y, then we establish that J∩X is a uo-band in X.

Uncountably infinite algebraic genericity and spaceability for sequence spaces

Let X be a topological vector space of complex-valued sequences and Y be a subset of X. We provide conditions for (X∖Y)∪{0} to contain uncountably infinitely many linearly independent dense vector subspaces of X. We also provide conditions for (X∖Y)∪{0} to contain uncountably infinitely many linearly independent closed infinite-dimensional vector subspaces of X. We apply these results to a chain of spaces containing the ℓp spaces.

Soft nodec spaces

<abstract><p>Following van Douwen, we call a soft topological space soft nodec if every soft nowhere dense subset of it is soft closed. This paper considers soft nodec spaces, which contain soft submaximal and soft door spaces. We investigate the basic properties and characterizations of soft nodec spaces. More precisely, we show that a soft nodec space can be written as a union of two disjoint soft closed soft dense (or soft open) soft nodec subspaces. Then, we study the behavior of soft nodec spaces under various operations, including the following: taking soft subspaces, soft products, soft topological sums, and images under specific soft functions with the support of appropriate counterexamples. Additionally, we show that the Krull dimension of a soft nodec soft $ T_{0} $-space is less than or equal to one. After that, we present some connections among soft nodec, soft strong nodec, and soft compact spaces. Finally, we successfully determine a condition under which the soft one-point compactification of a soft space is soft nodec if and only if the soft space is soft strong nodec.</p></abstract>

The Rough with the Smooth of the Light Cone String

The polynomials in the generators of a unitary representation of the Poincaré group constitute an algebra which maps the dense subspace D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal D$$\\end{document} of smooth, rapidly decreasing wavefunctions to itself. This mathematical result is highly welcome to physicists, who previously just assumed their algebraic treatment of unbounded operators be justified. The smoothness, however, has the side effect that a rough operator R, which does not map a dense subspace of D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal D$$\\end{document} to itself, has to be shown to allow for some other dense domain which is mapped to itself both by R and all generators. Otherwise their algebraic product, their concatenation, is not defined. Canonical quantization of the light cone string postulates operators -iX1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\ extrm{i}X^1$$\\end{document} and P-=(P0-Pz)/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^-=(P^0 - P^z)/2$$\\end{document} and as their commutator the multiplicative operator R=P1/(P0+Pz)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=P^1/(P^0 + P^z)$$\\end{document}. This is not smooth but rough on the negative z-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z-$$\\end{document}axis of massless momentum. Using only the commutation relations of Pm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^m$$\\end{document} with the generators -iMiz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\ extrm{i}M_{iz}$$\\end{document} of rotations in the Pi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^i$$\\end{document}-Pz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^{z}$$\\end{document}-plane we show that on massless states the operator R is inconsistent with a unitary representation of SO(D-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(D-1)$$\\end{document}. This makes the algebraic determination of the critical dimension, D=26\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=26$$\\end{document}, of the bosonic string meaningless: if the massless states of the light cone string admit R then they do not admit a unitary representation of the subgroup SO(D-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(D-1)$$\\end{document} of the Poincaré group. With analogous arguments we show: Massless multiplets are inconsistent with a translation group of the spatial momentum which is generated by a self-adjoint spatial position operator X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{X}$$\\end{document}.

A compact space K is Corson compact if and only if Cp(K) has a dense lc-scattered subspace

We establish that a compact space K is Corson compact if and only if it can be embedded in a topological group which is a W-space in the sense of Gruenhage. As a application of this result we show that K is Corson compact if and only if it embeds in Cp(X) for a Lindelöf scattered space X. We also prove that a zero-dimensional compact space K is Corson if and only if Cp(K,{0,1}) is a continuous image of a Lindelöf scattered space. Besides, a zero-dimensional countably compact space X is ω-monolithic if and only if Cp(X) has a dense functionally countable subspace.

Remark on ordered braid groups

We recover the Dehornoy order on the braid group [Formula: see text] from the tracial state on a cluster [Formula: see text]-algebra [Formula: see text] associated to the surface [Formula: see text] of genus [Formula: see text] with [Formula: see text] boundary components. It is proved that the space of left-ordering of the fundamental group [Formula: see text] is a totally disconnected dense subspace of the projective Teichmüller space [Formula: see text]. In particular, each left-ordering of [Formula: see text] defines the orbit of a Riemann surface [Formula: see text] under the geodesic flow on the space [Formula: see text].

Extensions of Functions with a Closed Graph and Quasi-continuous Functions with a Closed Graph from Dense Subspaces

In this paper, we prove some conditions for existence of an extension of a real (quasi-continuous) function with a closed graph defined on a given dense subset D of a topological space X to a (quasi-continuous) function with a closed graph on whole X.

On homotopically dense subspaces of the space of complete linked systems

On homotopically dense subspaces of the space of complete linked systems

An Exploration of the Reform of English Informatisation Teaching in Colleges and Universities Based on Deep Learning Model and Microteaching Mode

Abstract In this paper, we use the cross-layer connectivity of residual networks in deep learning to convert convolutional and fully connected layers into sparse connections and cluster sparse matrices into relatively dense subspaces. Extracted features are used to perform target class prediction and regression of target coordinates using a target detection algorithm to meet the demand for real-time target detection. The model's use resulted in a head-up rate of 83.57% in the classroom, with the least serious students at 0.8 and above. Deep learning technology can enhance students' learning experience in English classrooms by providing personalized learning and a deep learning environment.

Local/global existence analysis of fractional wave equations with exponential nonlinearity

In this paper, we concern with an exponential nonlinearity for a fractional wave equation in the whole space, we establish the local existence of solutions in a dense subspace of the Orlicz classification. Moreover, we obtain the global existence of solutions for small initial data in lower dimension 1≤d≤3. Our proofs base on the analyticity of Mittag-Leffler functions, the framework of prior estimates and the type of exponential nonlinearity.

A few characterizations of topological spaces with no infinite discrete subspace

We give several characteristic properties of FAC spaces, namely topological spaces with no infinite discrete subspace. The first one was obtained in 2019 by the first author, and states that every closed set is a finite union of irreducible closed subsets. The full result extends well-known characterizations of posets with no infinite antichain. One of them is that FAC spaces are, equivalently, topological spaces in which every closed set contains a dense Noetherian subspace, or spaces in which every Hausdorff subspace is finite, or in which no subspace has any infinite relatively Hausdorff subset. The latter comes with a nice min-max property, extending an observation of Erdős and Tarski in the case of posets: on spaces with no infinite relatively Hausdorff subset, the cardinalities of relatively Hausdorff subsets are bounded, and the least upper bound is also the least cardinality of a family of closed irreducible subsets that cover the space.

Ballistic transport for limit-periodic Schrödinger operators in one dimension

In this paper, we consider the transport properties of the class of limit-periodic continuum Schrödinger operators whose potentials are approximated exponentially quickly by a sequence of periodic functions. For such an operator H , and X_H(t) the Heisenberg evolution of the position operator, we show the limit of \frac{1}{t}X_H(t)\psi as t\to\infty exists and is nonzero for \psi\ne 0 belonging to a dense subspace of initial states which are sufficiently regular and of suitably rapid decay. This is viewed as a particularly strong form of ballistic transport, and this is the first time it has been proven in a continuum almost periodic non-periodic setting. In particular, this statement implies that for the initial states considered, the second moment grows quadratically in time.

On the density of CV0(X)⊗A in CV0(X,A)

Let X be a completely regular Hausdorff space and V a Nachbin family on X. For a locally convex algebra A, let CV0(X,A) be the algebra of all weighted vector-valued continuous functions which vanish at infinity with the topology given by the uniform seminorms induced by V. In this paper we present some sufficient conditions under which CV0(X)⊗A is a dense subspace of CV0(X,A) or isomorphic to it.

Smooth norms in dense subspaces of ℓp(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _p(\\Gamma )$$\\end{document} and operator ranges

For 1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p<\\infty $$\\end{document}, we prove that the dense subspace Yp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Y}_p$$\\end{document} of ℓp(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _p(\\Gamma )$$\\end{document} comprising all elements y such that y∈ℓq(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y \\in \\ell _q(\\Gamma )$$\\end{document} for some q∈(0,p)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q \\in (0,p)$$\\end{document} admits a C∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{\\infty }$$\\end{document}-smooth norm which locally depends on finitely many coordinates. Moreover, such a norm can be chosen as to approximate the ·p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left\\| \\cdot \\right\\| _p$$\\end{document}-norm. This provides examples of dense subspaces of ℓp(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _p(\\Gamma )$$\\end{document} with a smooth norm which have the maximal possible linear dimension and are not obtained as the linear span of a biorthogonal system. Moreover, when p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document} or Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in ℓ1(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1(\\Gamma )$$\\end{document} admits a C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}-smooth norm.