Epicompletion in Frames with Skeletal Maps,III: When Maps are Closed

In previous work it was shown that there is an epireflection ψ of the category of all compact normal, joinfit frames, with skeletal maps, in the full subcategory of frames which are also strongly projectable, and that ψ restricts to the epicompletion ε, which is the absolute reflection on compact regular frames. In the first part of this paper it is shown that ψ is a monoreflection and that the reflection map is, in fact, closed. Restricted to coherent frames and maps, ψ A can then be characterized as the least strongly projectable, coherent, normal, joinfit frame in which A can be embedded as a closed, coherent, and skeletal subframe. The second part discusses the role of the nucleus d in this context. On algebraic frames with coherent skeletal maps d becomes an epireflection. Further, it is shown that e = d · ψ epireflects the category of coherent, normal, joinfit frames, with coherent skeletal maps, in the subcategory of those frames which are also regular and strongly projectable, which are epicomplete. The action of e is not monoreflective.     

Epicompletion in Frames with Skeletal Maps,I: Compact Regular Frames

A frame homomorphism h : A ⟶ B is skeletal if x ^⊥⊥ = 1 in A implies that h(x)^⊥⊥ = 1 in B. It is shown that, in , the category of compact regular frames with skeletal maps, the subcategory , consisting of the frames in which every polar is complemented, coincides with the epicomplete objects in . Further, is the least epireflective subcategory, and, indeed, the target of the monoreflection which assigns to a compact regular frame A, the ideal frame ε A of , the boolean algebra of polars of A.      

Epicompletion in Frames with Skeletal Maps, II: Compact Normal Joinfit Frames

An epireflection ψ is constructed of the category $\mathfrak{KNArS}$ of compact normal joinfit frames, with skeletal maps, in the subcategory $\mathfrak{SPArS}$ consisting of strongly projectable $\mathfrak{KNArS}$ -objects. The construction is achieved via a pushout in the category $\mathfrak{FrmS}$ of frames with skeletal maps, and involves rather intimately the regular coreflection of the object to be reflected. Further, if the regular coreflection ρ is applied to the reflection map ψ _A :A?→?ψA one obtains the extension of ρA to its absolute.     

Saturation, Yosida Covers and Epicompleteness in Compact Normal Frames

In this article the frame-theoretic account of what is archimedean for order-algebraists, and semisimple for people who study commutative rings, deepens with the introduction of ${\mathcal{J}}$ -frames: compact normal frames that are join-generated by their saturated elements. Yosida frames are examples of these. In the category of ${\mathcal{J}}$ -frames with suitable skeletal morphisms, the strongly projectable frames are epicomplete, and thereby it is proved that the monoreflection in strongly projectable frames is the largest such. That is news, because it settles a problem that had occupied the first-named author for over five years. In compact normal Yosida frames the compact elements are saturated. When the reverse is true one gets the perfectly saturated frames: the frames of ideals Idl(E), when E is a compact regular frame. The assignment E?Idl(E) is then a functorial equivalence from compact regular frames to perfectly saturated frames, and the inverse equivalence is the saturation quotient. Inevitable are the Yosida covers (of a ${\mathcal{J}}$ -frame L): coherent, normal Yosida frames of the form Idl(F), with F ranging over certain bounded sublattices of the saturation SL of L. These Yosida frames cover L in the sense that each maps onto L densely and codensely. Modulo an equivalence, the Yosida covers of L form a poset with a top ${\mathcal{Y}} L$ , the latter being characterized as the only Yosida cover which is perfectly saturated. Viewed correctly, these Yosida covers provide, in a categorical setting, another (point-free) look at earlier accounts of coherent normal Yosida frames.     

Singular Archimedean lattice-ordered groups

W stands for the category of all archimedean l-groups with designated weak unit. The subcategory W _s of all groups with singular weak unit is analyzed as a full subcategory of W which is both epireflective and monocoreflective. A general technique for "contracting" monoreflections of a category A to a monocoreflective subcategory B is developed and then applied to W _s to show that: (i) the projectable hull in W _s is a monoreflection; (ii) essential hulls in W _s are formed by simply taking the lateral completion, and G is essentially closed in this category if and only if , where X is compact, Hausdorff and extremally disconnected; (iii) the maximum monoreflection on W _s , denoted , is obtained by contracting the maximum monoreflection on W, and G is epicomplete in W _s precisely when G is laterally -complete; (iv) the maximum essential reflection on W _s , denoted , is the contraction of the maximum essential reflection on W. Received January 22, 1997; accepted in final form June 11, 1998.     

Patch-generated Frames and Projectable Hulls

This article considers coherent frame homomorphisms h : L → M between coherent frames, which induce an isomorphism between the boolen frames of polars, with M projectable, and such that M is generated by h(L) and certain complemented elements of M. This abstracts the passage from a semiprime commutative ring with identity to its projectable hull. The frame theoretic setting is investigated thoroughly, first without any assumptions beyond the Zermelo–Fraenkel axioms of set theory, and, subsequently, assuming that algebraic frames are spatial. The culmination of this effort is the result that the spectrum of d-elements of M is obtained from that of L by refining the given hull–kernel topology to the patch topology. The second part of the article relates the projectable hull to the (von Neumann) regular hull, in a variety of contexts, including that of f-rings. For a uniformly complete f-algebra A, it is shown that the maximal ℓ-ideals of A that are traces of real maximal ideals of the regular hull HA are precisely the almost P-points of the space of maximal ℓ-ideals of A. For Bernhard Banaschewski, on the occasion of his 80th birthday.     

Irregular wavelet frames on<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ<Superscript>n</Superscript>)

In this paper, we present the conditions on dilation parameter {s _j}_j that ensure a discrete irregular wavelet system {s _j ^n/2ψ(s _j ·−bk)} _{j∈ℤ,k∈ℤ} ⁿ to be a frame on L²(ℝⁿ), and for the wavelet frame we consider the perturbations of translation parameter b and frame function ψ respectively.     

Compact Hausdorff Approach Frames

In this paper we provide a Heine–Borel type characterization for 0-compactness in approach spaces (Lowen 1997). Since this requires making use of the so-called regular function frame the most natural setting to develop this in is approach frames (Banaschewski 1999; Banaschewski et al., Acta Math Hung 115(3):183–196, 2007, Topology Appl 153:3059–3070, 2006). We then go on to characterize Hausdorffness for approach frames which allows us to study some fundamental properties of compact Hausdorff approach frames.     

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L ₂(ℝ ^d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H ^s (ℝ ^d ) and H ^−s (ℝ ^d ). In terms of masks for φ,ψ ¹,…,ψ ^L ∈H ^s (ℝ ^d ) and , simple sufficient conditions are given to ensure that (X ^s (φ;ψ ¹,…,ψ ^L ), forms a pair of dual wavelet frames in (H ^s (ℝ ^d ),H ^−s (ℝ ^d )), where

Decomposition of stochastic flows and Lyapunov exponents

Let φ _t be the stochastic flow of a stochastic differential equation on a compact Riemannian manifold M. Fix a point m∈M and an orthonormal frame u at m, we will show that there is a unique decomposition φ _t = ξ _t ψ _t such that ξ _t is isometric, ψ _t fixes m and Dψ _t (u) = us _t , where s _t is an upper triangular matrix. We will also establish some convergence properties in connection with the Lyapunov exponents and the decomposition Dφ _t (u) = u _t s _t with u _t being an orthonormal frame. As an application, we can show that ψ_t preserves the directions in which the tangent vectors at m are dilated at fixed exponential rates. Received: 19 November 1998 / Revised version: 1 October 1999 / Published online: 14 June 2000     

Frames and their associated $\emph{H}_{{\kern-2pt}\emph{F}}^{\emph{p}}$-subspaces

Given a frame F = {f _j } for a separable Hilbert space H, we introduce the linear subspace H^p_FH^{p}_{F} of H consisting of elements whose frame coefficient sequences belong to the ℓ ^p -space, where 1 ≤ p < 2. Our focus is on the general theory of these spaces, and we investigate different aspects of these spaces in relation to reconstructions, p-frames, realizations and dilations. In particular we show that for closed linear subspaces of H, only finite dimensional ones can be realized as H^p_FH^{p}_{F}-spaces for some frame F. We also prove that with a mild decay condition on the frame F the frame expansion of any element in H_F^pH_{F}^{p} converges in both the Hilbert space norm and the ||·|| _{F, p} -norm which is induced by the ℓ ^p -norm.     

Constructing pairs of dual bandlimited framelets with desired time localization

For sufficiently small translation parameters, we prove that any bandlimited function ψ, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame with a dual frame also having the wavelet structure. This dual frame is generated by a finite linear combination of dilations of ψ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction is based on characterizing equations for dual wavelet frames and relies on a technical condition. We exhibit a general class of function satisfying this condition; in particular, we construct piecewise polynomial functions satisfying the condition.      

Cartan–Laptev method in the theory of multidimensional three-webs

We show how the Cartan–Laptev method that generalizes Elie Cartan’s method of external forms and moving frames is applied to the study of closed G-structures defined by multidimensional three-webs formed on a C ^s -smooth manifold of dimension 2r, r ≥ 1, s ≥ 3, by a triple of foliations of codimension r. We say that a tensor T belonging to a differential-geometric object of order s of a three-web W is closed if it can be expressed in terms of components of objects of lower order s. We find all closed tensors of a three-web and the geometric sense of one of relations connecting three-web tensors. We also point out some sufficient conditions for the web to have a closed G-structure. It follows from our results that the G-structure associated with a hexagonal three-web W is a closed G-structure of class 4. It is proved that basic tensors of a three-web W belonging to a differential-geometric object of order s of the web can be expressed in terms of an s-jet of the canonical expansion of its coordinate loop, and conversely. This implies that the canonical expansion of every coordinate loop of a three-web W with closed G-structure of class s is completely defined by an s-jet of this expansion. We also consider webs with one-digit identities of kth order in their coordinate loops and find the conditions for these webs to have the closed G-structure.     

Von Neumann-Jordan Constants of Absolute Normalized Norms on C^n

In this note, we give some estimations of the Von Neumann-Jordan constant C _{N J} (∥·∥_ψ) of Banach space (ℂ ⁿ , ∥·∥_ψ), where ∥·∥_ψ is the absolute normalized norm on ℂ ⁿ given by function ψ. In the case where ψ and φ are comparable, n=2 and C _{N J} (∥·∥_ψ)=1, we obtain a formula of computing C _{N J} (∥·∥_ψ). Our results generalize some results due to Saito and others. Received May 11, 2002, Accepted November 20, 2002 This work is partly supported by NNSF of China (No. 19771056)     

Some characterization results based on the conditional expectation of a function of non-adjacent order statistic (record value)

In this paper, we attempt to characterize a distribution by means ofE[ψ(X _k +s:n)|X _k:n =z]g(z), under some mild conditions on ψ(·) andg(·). An explicit result is provided in the case ofs=1 and a uniqueness result is proved in the case ofs=2. For the general case, an expression is provided for the conditional expectation. Similar results are proved for the record values, both in the continuous as well as in the discrete case (weak records).     

On the equivalence of measures on loop space

Let K be a simply-connected compact Lie Group equipped with an Ad _K -invariant inner product on the Lie Algebra ?, of K. Given this data, there is a well known left invariant “H ¹-Riemannian structure” on L(K) (the infinite dimensional group of continuous based loops in K), as well as a heat kernel ν_T(k ₀, ·) associated with the Laplace-Beltrami operator on L(K). Here T > 0, k ₀∈L(K), and ν _T (k ₀, ·) is a certain probability measure on L(K). In this paper we show that ν₁(e,·) is equivalent to Pinned Wiener Measure on K on ? _s0 ≡<x _t : t∈ [0, s ₀]> (the σ-algebra generated by truncated loops up to “time”s ₀). Recevied: 9 September 1999 / Revised version: 13 March 2000 / Published online: 22 November 2000     

Probabilistic Representations of Solutions of the Forward Equations

In this paper we prove a stochastic representation for solutions of the evolution equation

A note on weakly Lindelöf frames

The notion of σ^?-properness of a subset of a frame is introduced. Using this notion, we give necessary and su?cient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ?? is provided. The closed ideals of ?? equipped with the uniform topology are applied to describe weakly Lindelöf frames.     

A structure theorem of Dirac-harmonic maps between spheres

For an arbitrary Dirac-harmonic map (φ,ψ) between compact oriented Riemannian surfaces, we shall study the zeros of |ψ|. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of |ψ| and the genus of M and N. On the basis, we could clarify all of non-trivial Dirac-harmonic maps from S ² to S ².     

Strong Nearness Frames Having Enough Cauchy Filters

We prove results establishing sufficient conditions for the sum of two nearness frames to have enough Cauchy filters. From these results and the fact that, in the category of strong nearness frames having enough Cauchy filters and uniform frame maps, complete spatial frames form a coreflective subcategory, follow a variety of results where the open-sets contravariant functor from topological spaces to frames transforms products into sums and inverse limits into direct limits.