A characterization theorem for tilings having countably many singular points

LetT be a tiling of the plane. At most countably many points of U{bdry T T inT} fail to lie in a nondegenerate edge ofT if and only ifT has at most countably many singular points. The result fails without the requirement that the edges be nondegenerate. Moreover, countably many cannot be replaced by finitely many in the theorem.     

Spaces of functions with countably many discontinuities

Let Γ be a Polish space and let K be a separable and pointwise compact set of functions on Γ. Assume further that each function in K has only countably many discontinuities. It is proved that admits a -lower semicontinuous and locally uniformly rotund norm, equivalent to the supremum norm. A slightly more general result is shown and a related conjecture is stated. The second author has been supported by BFM2003-07540/MATE, Ministerio de Ciencia y Tecnología MCIT y FEDER (Spain). The third author has been supported by Fundación Séneca CARM: 00690/PI/04 and by BFM2002-01719 del MCIT.     

Covering functions by countably many functions from some families

Let $ \mathcal{A} $ be a nonempty family of functions from $ \mathbb{R} $ to $ \mathbb{R} $ . A function $ f:\mathbb{R}\to \mathbb{R} $ is said to be strongly countably $ \mathcal{A} $ -function if there is a sequence (f _n ) of functions from $ \mathcal{A} $ such that $ \mathrm{Gr}(f)\subset {\cup_n}\mathrm{Gr}\left( {{f_n}} \right) $ (Gr(f) denotes the graph of f). If $ \mathcal{A} $ is the family of all continuous functions, the strongly countable $ \mathcal{A} $ -functions are called strongly countably continuous and were investigated in [Z. Grande and A. Fatz-Grupka, On countably continuous functions, Tatra Mt. Math. Publ., 28:57–63, 2004], [G. Horbaczewska, On strongly countably continuous functions, Tatra Mt. Math. Publ., 42:81–86, 2009], and [T.A. Natkaniec, On additive countably continuous functions, Publ. Math., 79(1–2):1–6, 2011]. In this article, we prove that the families $ \mathcal{A}\left( \mathbb{R} \right) $ of all strongly countably $ \mathcal{A} $ -functions are closed with respect to some operations in dependence of analogous properties of the families $ \mathcal{A} $ , and, in particular, we show some properties of strongly countably differentiable functions, strongly countably approximately continuous functions, and strongly countably quasi-continuous functions.     

Correction and representation theorems for functions of countably many variables

Some similarities of the well-known theorems on correction and representation of functions of one and several variables are proved for functions of countably many variables.     

On the corona theorem for almost periodic functions

LetAP ⁺ (R ⁿ ) denote the Banach algebra of all continuous almost periodic functions onR ⁿ whose Bohr-Fourier spectrum is contained in an additive semi-group [0, ) ⁿ . We show that the maximal ideal space ofAP ⁺ (R ⁿ ) may have a nonempty corona and we characterize all for which the corona is empty. Analogous results are established for algebras of almost periodic functions with absolutely convergent Fourier series.     

A matricial corona theorem

We show that a usual corona-type theorem on a space of functions automatically extends to a matrix version.

     

Sarason interpolation and Toeplitz corona theorem for almost periodic matrix functions

Sarason interpolation and Toeplitz corona problems are studied for almost periodic matrix functions. Recent results on almost periodic factorization and related generalized Toeplitz operators are the main tools in the study.Supported in part by NSF Grant DMS 9500912Supported in part by NATO Collaborative Research Grant 950332Supported by NSF Grant DMS 9500924     

A matricial corona theorem II

We extend the ``matricial corona theorem' of M. Andersson to general algebras of functions which satisfy a corona theorem.

     

Rings with countably many direct summands

We consider commutative monoid amalgams having as their cores a conical cancellative monoid. In this context we present sufficient conditions for the amalgam to be strongly embeddable and for some properties of the factors to be transmitted to the amalgamated free product.     

Markets with countably infinitely many traders

In this paper we will show how to construct, in a canonical way, a continuous economy in the sense of R. J. Aumann [1]. This construction is based on W. Hildenbrand's [3] definition of “pure competition” for an exchange economy with countably infinitely many participants.     

The problem of fair division for countably many participants

We show the existence of a fair division for countably many participants and that the related cooperative game is totally balanced.     

Linear-fractional branching processes with countably many types

We study multi-type Bienaymé–Galton–Watson processes with linear-fractional reproduction laws using various analytical tools like the contour process, spinal representation, Perron–Frobenius theorem for countable matrices, and renewal theory. For this special class of branching processes with countably many types we present a transparent criterion for R$R$-positive recurrence with respect to the type space. This criterion appeals to the Malthusian parameter and the mean age at childbearing of the associated linear-fractional Crump-Mode-Jagers process.     

Decomposing uncountable squares to countably many chains

We construct an ordered set I of cardinality $\text{N}$₁, such that its square is the union of $\text{N}$₀ chains (in the natural partial order).     

Extending Baire Property by countably many sets

We prove that if ZFC is consistent so is ZFC + ``for any sequence of subsets of a Polish space there exists a separable metrizable topology on with , and Borel in for all .' This is a category analogue of a theorem of Carlson on the possibility of extending Lebesgue measure to any countable collection of sets. A uniform argument is presented, which gives a new proof of the latter as well.

Some consequences of these extension properties are also studied.

     

The invariance of domain theorem for countably 1-set-contractive mappings

The purpose of this paper is to extend the invariance of domain theorem to a large class of countably 1-γ-contractive maps, by using homotopy theory and degree theory for countably 1-γ-contractive maps.     

The generalized corona theorem

The results of this paper have been presented by the second author at the joint AMS-DMV Meeting 1993 in Heidelberg (Germany)