The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model ...

Every nontrivial countably productive coreflective subcategory of topological linear spaces is κ-productive for a large cardinal κ (see [10]). Unlike that case, in uniform spaces for every infinite regular cardinal κ, there are coreflective subcategories that are κ-productive and not κ-productive (see [8]). From certain points of view, the category of topological groups lies in between those ca...

Now it is obvious from the usual definitions that an inaccessible κ is subtle iff every κ-tree T satisfies STP(T ), for which we shall just write κ-STP, iff the complete binary tree 2<κ satisfies STP(2<κ), and similarly for ineffability (and one can take this as a definition if unfamiliar with the concepts). By [Mit73] one can collapse a weakly compact (a Mahlo) cardinal onto ω2 such that in th...

We study the determinacy of the game Gκ(A) introduced in [FuKoShe] for uncountable regular κ and several classes of partial orderings A. Among trees or Boolean algebras, we can always find an A such that Gκ(A) is undetermined. For the class of linear orders, the existence of such A depends on the size of κ. In particular we obtain a characterization of κ = κ in terms of determinacy of the game ...

We study the determinacy of the game Gκ(A) introduced in [FuKoShe] for uncountable regular κ and several classes of partial orderings A. Among trees or Boolean algebras, we can always find an A such that Gκ(A) is undetermined. For the class of linear orders, the existence of such A depends on the size of κ. In particular we obtain a characterization of κ = κ in terms of determinacy of the game ...

For a regular κ, a partial ordering P is said to have the κ Freese-Nation property (the κ-FN for short) if there is a mapping (κ-FN mapping) f : P → [P ] such that for any p, q ∈ P if p ≤ q then there is r ∈ f(p) ∩ f(q) such that p ≤ r ≤ q. Freese and Nation [5] used the א0-FN in a characterization of projective lattices and asked if this property alone already characterizes the projectiveness....

A new four elements compact antenna array is presented and discussed to achieve enhanced phase resolution without sacrificing the array output power. This structure inspired by the Ormia Ochracea’s coupled ears. The analogy between this insect acute directional hearing capabilities and the electrically compact antenna array is used to enhance the array sensitivity to direction of arrival estima...

We show that if κ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size κ is invariantly universal. This means that for every analytic quasi-order on the generalized Cantor space 2 there is an Lκ+κsentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bireducible (and, in fact, classwise Borel isomorphic) to R...

Assuming GCH $\mathsf {GCH}$ , we show that generalized eventually narrow sequences on a strongly inaccessible cardinal κ are preserved under one step iteration of the Hechler forcing for adding dominating κ-real. Moreover, if is unfoldable, 2 = + $2^\kappa =\kappa ^+$ and λ regular such < $\kappa ^+ \lambda$ then there set generic extension in which s ( ) b c $\mathfrak {s}(\kappa \kappa \math...