Blown-up Čech cohomology and Cartan’s Theorem B on real algebraic varieties

We introduce a concept of blown-up Čech cohomology for coherent sheaves of homological dimension [Formula: see text] and some quasi-coherent sheaves on a nonsingular real affine variety. Its construction involves a directed set of multi-blowups. We establish, in particular, long exact cohomology sequence and Cartan’s Theorem B. Finally, some applications are provided, including universal solution to the first Cousin problem (after blowing up).

INTERSECTIONS OF TRANSLATION OF A CLASS OF SIERPINSKI CARPETS

For [Formula: see text], a middle-[Formula: see text] Sierpinski carpet [Formula: see text] is defined as the self-similar set generated by the iterated function system (IFS) [Formula: see text], where [Formula: see text] is defined by [Formula: see text] Here, [Formula: see text]. In this paper, for [Formula: see text], we investigated the equivalent characterizations of the intersection [Formula: see text] being a generalized Moran set. Furthermore, under some conditions, we show that [Formula: see text] can be represented as a graph-directed set satisfying the open set condition (OSC), and then the Hausdorff dimension can be explicitly calculated.

Pringsheim convergence of double sequences for uniform boundedness principle

The set of all pairs of positive integers is considered as a directed set under the direction: [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. This directed set is used for Pringsheim-type convergence of double sequences. Consequences of uniform boundedness principle through double sequences are derived in this paper.

A net convergence for Schauder double bases

The set of all pairs of natural numbers is considered as a directed set under the direction: [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. This directed set is used to study convergence of a double series in a sense of Pringsheim and to introduce double bases in topological vector spaces. An introductory study on double bases is presented.

Summable Family in a Commutative Group

Summary Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [22], [7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space [30], a real normed linear space [29] and a linear topological space [14] with the concept of the limit of an image filter [16]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commutative group (“additive notation” in [17]), using the notion of filters.

Real Dimension Groups

Dimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like; for instance, it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In an appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.

THE HAUSDORFF DIMENSION OF A GRAPH-DIRECTED SET WHOSE UNDERLYING MULTIGRAPH IS A CARTESIAN PRODUCT OR A TENSOR PRODUCT OF MULTIGRAPHS

In this paper, we compute the Hausdorff dimension of a graph-directed set when the underlying multigraph is a Cartesian product or a tensor product of several multigraphs. We give explicit formulas in terms of the eigenvalues of the graph and the similarity ratios used with each graph.

The bounded proper forcing axiom

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ1 many maximal antichains of a proper forcing notion, each of size ℵ1, there is a directed set meeting all these antichains.A regular cardinal κ is called ∑1-reflecting, if for any regular cardinal χ, for all formulas φ, “H(χ) ⊨ ‘φ’” implies “∃δ < κ, H(δ) ⊨ ‘φ’”.We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the bounded proper forcing axiom is exactly the existence of a ∑1-reflecting cardinal (which is less than the existence of a Mahlo cardinal).We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure.

Some remarks on pramarts and mils

Let F be a Banach space, (ω, ℱ, P) a fixed probability space, D a directed set filtering to the right with the order ≤, and (ℱt, D) a stochastic basis of ℱ, i.e. (ℱt, D) is an increasing family of sub-σ-algebras of ℱ:ℱs ⊂ for any s,t ε D and s≤t. Throughout this paper, (Xt) is an F-valued, (ℱt)-adapted sequence, i.e. Xt, is ℱt-measurable, t ε D. We also assume that Xt, ∈ L1, i.e. ∫ ∥Xt∥ <∞. We use I(H) to denote the indicator function of an event H. Let ∞ be a such element: t <∞, t ∈ D, = D ∪ ∞, and ℱ∞ = σ. A stopping time is a map τ:Ω→ such that (τ<t) ∈ ℱt, t ∈ D. A stopping time τ is called simple (countable) if it takes finitely (countably) many values in D(). Let T and Tc be the sets of simple and countable stopping times respectively and Tf = {τ ∈ Tc: τ<∞ a.s.}. Clearly, (T, <) and (Tf, <) are directed sets filtering to the right. For τ ∈ Tc, letand= {(Xt): there is σ∈ Tf such that ∫(ι<∞) ∥Xι∥ < ∞, σ ≤ τ ∈ Tc},= {(Xt):(Xι, ι ∈ T) converges stochastically (i.e. in probability) in the norm topology},ℰ = {(Xt):(Xι, ι ∈ T) converges essentially in the norm topology}.