On the strong amalgamation of Boolean algebras

This paper presents a proof that Boolean algebras have the strong amalgamation property which does not use any choice principle, based on a feature of their binary coproducts which is derived by means of a construction from pointfree topology.     

Pointfree Spectra of Riesz Spaces

One of the best ways of studying ordered algebraic structures is through their spectra. The three well-known spectra usually considered are the Brumfiel, Keimel, and the maximal spectra. The pointfree versions of these spectra were studied by B. Banaschewski for f-rings. Here, we give the pointfree versions of the Keimel and the maximal spectra for Riesz spaces. Moreover, we briefly mention how one can use the results of this paper to give a pointfree version of the Kakutani duality for Riesz spaces.     

On Compactness of Frames

We define the notions of extension-closed and nearly closed sublocales of frames and present, amongst other things, a pointfree and choice-free proof of a result of Harris characterizing extension-closed subspaces. We also characterize compact frames in terms of the latter form of closedness.     

Lower and upper regularizations of frame semicontinuous real functions

As discovered recently, Li and Wang’s 1997 treatment of semicontinuity for frames does not faithfully reflect the classical concept. In this paper we continue our study of semicontinuity in the pointfree setting. We define the pointfree concepts of lower and upper regularizations of frame semicontinuous real functions. We present characterizations of extremally disconnected frames in terms of these regularizations that allow us to reprove, in particular, the insertion and extension type characterizations of extremally disconnected frames due to Y.-M. Li and Z.-H. Li [Algebra Universalis 44 (2000), 271–281] in the right semicontinuity context. It turns out that the proof of the insertion theorem becomes very easy after having established a number of basic results regarding the regularizations. Notably, our extension theorem is a much strengthened version of Li and Li's result and it is proved without making use of the insertion theorem. The first and second named authors acknowledge financial support from the Ministry of Education and Science of Spain and FEDER under grant MTM2006-14925-C02-02. The first named author also acknowledges financial support from the University of the Basque Country under grant UPV05/101. The third named author acknowledges financial support from the Centre of Mathematics of the University of Coimbra/FCT.     

Tightness relative to some (co)reflections in topology

We address what might be termed the reverse reflection problem: given a monoreflection from a category A onto a subcategory B, when is a given object B ∈ B the reflection of a proper subobject? We start with a well known specific instance of this problem, namely the fact that a compact metric space is never the ?ech-Stone compactification of a proper subspace. We show that this holds also in the pointfree setting, i.e., that a compact metrizable locale is never the ?ech-Stone compactification of a proper sublocale. This is a stronger result than the classical one, but not because of an increase in scope; after all, assuming weak choice prin­ciples, every compact regular locale is the topology of a compact Hausdorff space. The increased strength derives from the conclusion, for in general a space has many more sublocales than subspaces. We then extend the analysis from metric locales to the broader class of perfectly normal locales, i.e., those whose frame of open sets consists entirely of cozero elements. We include a second proof of these results which is purely algebraic in character.

At the opposite extreme from these results, we show that an extremally disconnected locale is a compactification of each of its dense sublocales. Finally, we analyze the same phenomena, also in the pointfree setting, for the 0-dimensional compact reflec­tion and for the Lindelöf reflection.     

A new look at localic interpolation theorems

This paper presents a new treatment of the localic Katětov-Tong interpolation theorem, based on an analysis of special properties of normal frames, which shows that it does not hold in full generality. Besides giving us the conditions under which the localic Katětov-Tong interpolation theorem holds, this approach leads to a especially transparent and succinct proof of it. It is also shown that this pointfree extension of Katětov-Tong theorem still covers the localic versions of Urysohn's Lemma and Tietze's Extension Theorem.     

On the parallel between normality and extremal disconnectedness

Several familiar results about normal and extremally disconnected (classical or pointfree) spaces shape the idea that the two notions are somehow dual to each other and can therefore be studied in parallel. This paper investigates the source of this ‘duality’ and shows that each pair of parallel results can be framed by the ‘same’ proof. The key tools for this purpose are relative notions of normality, extremal disconnectedness, semicontinuity and continuity (with respect to a fixed class of complemented sublocales of the given locale) that bring and extend to locale theory a variety of well-known classical variants of normality and upper and lower semicontinuities in an illuminating unified manner. This approach allows us to unify under a single localic proof all classical insertion, as well as their corresponding extension results.     

Monotonicity and finite element methods for hyperbolic initial value problems

A well known theórem about super- and subfunctions for the solution of hyperbolic initial value problems constructs differentiable functions as upper and lower bounds (see Walter [1], 21 XIII). The proof can be done by transforming the differential equation problem into a set of integral equations, using the monotonicity-properties of the arising integral operators. This proof needs an integral representation for twice differentiable functions. It is shown that this proceeding can be generalized to get upper and lower bounds in terms of finite element functions. To do this, we give an integral representation for continuous, piecewise differentiable functions, including the discontinuities of their derivatives. Then the generalization of the classical proof yields interface conditions for the finite element functions. Finally, it is demonstrated how to realize numerically these conditions.     

Fundamental results for pointfree convex geometry

Inspired by locale theory, we propose “pointfree convex geometry”. We introduce the notion of convexity algebra as a pointfree convexity space. There are two notions of a point for convexity algebra: one is a chain-prime meet-complete filter and the other is a maximal meet-complete filter. In this paper we show the following: (1) the former notion of a point induces a dual equivalence between the category of “spatial” convexity algebras and the category of “sober” convexity spaces as well as a dual adjunction between the category of convexity algebras and the category of convexity spaces; (2) the latter notion of point induces a dual equivalence between the category of “m-spatial” convexity algebras and the category of “m-sober” convexity spaces. We finally argue that the former notion of a point is more useful than the latter one from a category theoretic point of view and that the former notion of a point actually represents a polytope (or generic point) and the latter notion of a point properly represents a point. We also remark on the close relationships between pointfree convex geometry and domain theory.     

Choiceless, Pointless, but not Useless: Dualities for Preframes

We provide the appropriate common ‘(pre)framework’ for various central results of domain theory and topology, like the Lawson duality of continuous domains, the Hofmann–Lawson duality between continuous frames and locally compact sober spaces, the Hofmann–Mislove theorems about continuous semilattices of compact saturated sets, or the theory of stably continuous frames and their topological manifestations. Suitable objects for the pointfree approach are quasiframes, i.e., up-complete meet-semilattices with top, and preframes, i.e., meet-continuous quasiframes. We introduce the pointfree notion of locally compact well-filtered preframes, show that they are just the continuous preframes (using a slightly modified definition of continuity) and establish several natural dualities for the involved categories. Moreover, we obtain various characterizations of preframes having duality. Our results hold in ZF set theory without any choice principles.      

On Point-finiteness in Pointfree Topology

In pointfree topology, the point-finite covers introduced by Dowker and Strauss do not behave similarly to their classical counterparts with respect to tran- sitive quasi-uniformities, contrarily to what happens with other familiar types of interior-preserving covers. The purpose of this paper is to remedy this by modifying the definition of Dowker and Strauss. We present arguments to justify that this modification turns out to be the right pointfree definition of point-finiteness. Along the way we place point-finite covers among the classes of interior-preserving and closure-preserving families of covers that are relevant for the theory of (transitive) quasi-uniformities, completing the study initiated with Ferreira and Picado, Kyungpook Math. J., 44: 415–442, 2004.     

A note on precompact uniform frames

Precompactness or total boundedness for uniform frames is usually distinguished by a cover approach. In this note, we provide alternate characterizations of precompact uniform frames. In particular, we formulate pointfree filter analogues of various classical topological results on precompactness. We also revisit the notion of convergence and clustering of filters in a frame and introduce weakly Cauchy filters and strong Cauchy completeness in the setting of uniform frames.     

On Strong Inclusions and Asymmetric Proximities in Frames

The strong inclusion, a specific type of subrelation of the order of a lattice with pseudocomplements, has been used in the concrete case of the lattice of open sets in topology for an expedient definition of proximity, and allowed for a natural pointfree extension of this concept. A modification of a strong inclusion for biframes then provided a pointfree model also for the non-symmetric variant. In this paper we show that a strong inclusion can be non-symmetrically modified to work directly on frames, without prior assumption of a biframe structure. The category of quasi-proximal frames thus obtained is shown to be concretely isomorphic with the biframe based one, and shown to be related to that of quasi-uniform frames in a full analogy with the symmetric case.     

Strong Cauchy completeness in uniform frames

We show that for uniform frames, with the underlying frame being Boolean, uniform paracompactness and strong Cauchy completeness are equivalent conditions. Certain aspects of uniform paracompactness are also considered. We then introduce the pointfree notion of preparacompactness and show that the completion of a preparacompact uniform frame is strongly Cauchy complete. We also formulate pointfree filter characterizations of Lindelöfness for regular frames in analogy to their classical topological counterparts.     

关于代数多项式的一个积分表示式

研究了代数多项式导数的Bernstein不等式和Markov不等式．通过代数多项式导数的一个积分表示式,给出这两个著名不等式以及它们的离散形式的证明．     

Some Special Tauberian Theorems for Fourier Type Integrals and Eigenfunction Expansions of Sturm-Liouville Equations on the Whole Line

We offer a new proof of a special Tauberian theorem for Fourier type integrals. This Tauberian theorem was already considered by us in the papers [1] and [2]. The idea of our initial proof was simple, but the details were complicated because we used Bochner's definition of generalized Fourier transform for functions of polynomial growth. In the present paper we work with L. Schwartz's generalization. This leads to significant simplification. The paper consists of six sections. In Section 1 we establish an integral representation of functions of polynomial growth (subjected to some Tauberian conditions), in Section 2 we prove our main Tauberian theorems (Theorems 2.1 and 2.2.), using the integral representation of Section 1, in Section 3 we study the asymptotic behavior of M. Riesz's means of functions of polynomial growth, in Sections 4 and 5 we apply our Tauberian theorems to the problem of equiconvergence of eigenfunction expansions of Sturm-Liouville equations and expansion in ordinary Fourier integrals, and in Section 6 we compare our general equiconvergence theorems of Sections 4 and 5 with the well known theorems on eigenfunction expansions in classical orthogonal polynomials. In some sense this paper is a re-made survey of our results obtained during the period 1953-58. Another proof of our Tauberian theorem and some generalization can be found in the papers [3] and [4].     

A NOTE ON COMPLETE REGULARITY AND NORMALITY

Abstract

In this paper we introduce the notions of uniform complete regularity and uniform normality for nearness frames with the view of obtaining new and pointfree proofs of some known topological results.     

二阶Volterra系统对平稳正态激励的统计响应的表示定理的数学证明

1974年,Neal根据Kac和Siegert的思想,给出了一个在电子工程、海洋工程、建筑工程、航空工程、自动控制的随机振动中有重要应用的二阶Volterra非线性系统对平稳正态输入的统计响应的表示定理.1984年,Naess对此定理又给出了一个数学证明.经过研究后发现,他们对定理条件的叙述都是模糊的,而且其数学证明都是有问题的.本文重新讨论了这个表示定理,给出了明确的定理条件及严格的数学证明,为它的广泛应用奠定了理论基础.     

On Weak Solutions of Stochastic Differential Equations

A new proof of existence of weak solutions to stochastic differential equations with continuous coefficients based on ideas from infinite-dimensional stochastic analysis is presented. The proof is fairly elementary, in particular, neither theorems on representation of martingales by stochastic integrals nor results on almost sure representation for tight sequences of random variables are needed.