Linear orderings and powers of characterizable cardinals

We say that a countable model M completely characterizes an infinite cardinal κ, if the Scott sentence of M has a model in cardinality κ, but no models in cardinality κ⁺. If a structure M completely characterizes κ, κ is called characterizable. In this paper, we concern ourselves with cardinals that are characterizable by linearly ordered structures (cf. Definition 2.1).Under the assumption of GCH, Malitz completely resolved the problem by showing that κ is characterizable if and only if κ=ℵ_α, for some α<ω₁ (cf. Malitz (1968) [7] and Baumgartner (1974) [1]). Our results concern the case where GCH fails.From Hjorth (2002) [3], we can deduce that if κ is characterizable, then κ⁺ is characterizable by a densely ordered structure (see Theorem 2.4 and Corollary 2.5).We show that if κ is homogeneously characterizable (cf. Definition 2.2), then κ is characterizable by a densely ordered structure, while the converse fails (Theorem 2.3).The main theorems are (1) If κ>2^λ is a characterizable cardinal, λ is characterizable by a densely ordered structure and λ is the least cardinal such that κ^λ>κ, then κ^λ is also characterizable (Theorem 5.4) and (2) if ℵ_α and κ^ℵ_α are characterizable cardinals, then the same is true for κ^ℵ_α+β, for all countable β (Theorem 5.5).Combining these two theorems we get that if κ>2^ℵ_α is a characterizable cardinal, ℵ_α is characterizable by a densely ordered structure and ℵ_α is the least cardinal such that κ^ℵ_α>κ, then for all β<α+ω₁, κ^ℵ_β is characterizable (Theorem 5.7). Also if κ is a characterizable cardinal, then κ^ℵ_α is characterizable, for all countable α (Corollary 5.6). This answers a question of the author in Souldatos (submitted for publication) [8].     

Cauchy transforms of self-similar measures: the Laurent coefficients

The Cauchy transform of a measure has been used to study the analytic capacity and uniform rectifiability of subsets in . Recently, Lund et al. (Experiment. Math. 7 (1998) 177) have initiated the study of such transform F of self-similar measure. In this and the forecoming papers (Starlikeness and the Cauchy transform of some self-similar measures, in preparation; The Cauchy transform on the Sierpinski gasket, in preparation), we study the analytic and geometric behavior as well as the fractal behavior of the transform F. The main concentration here is on the Laurent coefficients {a_n}_n=0^∞ of F. We give asymptotic formulas for {a_n}_n=0^∞ and for F^(k)(z) near the support of μ, hence the precise growth rates on |a_n| and |F^(k)| are determined. These formulas are connected with some multiplicative periodic functions, which reflect the self-similarity of μ and K. As a by-product, we also discover new identities of certain infinite products and series.     

Selective screenability in topological groups

We examine the selective screenability property in topological groups. In the metrizable case we also give characterizations of Sc(Onbd,O) and Smirnov-Sc(Onbd,O) in terms of the Haver property and finitary Haver property respectively relative to left-invariant metrics. We prove theorems stating conditions under which Sc(Onbd,O) is preserved by products. Among metrizable groups we characterize the countable dimensional ones by a natural game.     

When does the Haver property imply selective screenability?

We point out that in metric spaces Haver's property is not equivalent to the property introduced by Addis and Gresham. We prove that they are equal when the space has the Hurewicz property. We prove several results about the preservation of Haver's property in products. We show that if a separable metric space has the Haver property, and the nth power has the Hurewicz property, then the nth power has the Addis-Gresham property. R. Pol showed earlier that this is not the case when the Hurewicz property is replaced by the weaker Menger property. We introduce new classes of weakly infinite dimensional spaces.     

Abstract elementary classes stable in ℵ0

We study abstract elementary classes (AECs) that, in ${\mathrm{?}}_0$, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such classes exhibit superstable-like behavior at ${\mathrm{?}}_0$. More precisely, there is a superlimit model of cardinality ${\mathrm{?}}_0$ and the class generated by this superlimit has a type-full good ${\mathrm{?}}_0$-frame (a local notion of nonforking independence) and a superlimit model of cardinality ${\mathrm{?}}_1$. We also give a supersimplicity condition under which the locality hypothesis follows from the rest.     

Strong γ-sets and other singular spaces

Whereas the Gerlits-Nagy γ property is strictly weaker than the Galvin-Miller strong γ property, the corresponding strong notions for the Menger, Hurewicz, Rothberger, Gerlits-Nagy (∗), Arkhangel'ski?ˇ and Sakai properties are equivalent to the original ones. The main result is that almost each of these properties admits the game theoretic characterization suggested by the stronger notion. We also solve a related problem of Ko?inac and Scheepers, and answer a question of Iliadis.     

Internal sizes in μ-abstract elementary classes

Working in the context of μ-abstract elementary classes (μ-AECs)—or, equivalently, accessible categories with all morphisms monomorphisms—we examine the two natural notions of size that occur, namely cardinality of underlying sets and internal size. The latter, purely category-theoretic, notion generalizes e.g. density character in complete metric spaces and cardinality of orthogonal bases in Hilbert spaces. We consider the relationship between these notions under mild set-theoretic hypotheses, including weakenings of the singular cardinal hypothesis. We also establish preliminary results on the existence and categoricity spectra of μ-AECs, including specific examples showing dramatic failures of the eventual categoricity conjecture (with categoricity defined using cardinality) in μ-AECs.     

An arithmetical view to first-order logic

A value space is a topological algebra B equipped with a non-empty family of continuous quantifiers :B^∗→B. We will describe first-order logic on the basis of B. Operations of B are used as connectives and its relations are used to define statements. We prove under some normality conditions on the value space that any theory in the new setting can be represented by a classical first-order theory.     

Computably categorical Boolean algebras enriched by ideals and atoms

We describe computably categorical Boolean algebras whose language is enriched by one-place predicates that distinguish a finite set of ideals and atoms with respect to some ideals in this set.     

Cosmic dimensions

Martin's Axiom for σ-centered partial orders implies that there is a cosmic space with non-coinciding dimensions.     

Geometric stability theory for μ-structures

We introduce a notion of μ-structures which are certain locally compact group actions and prove some counterparts of results on Polish structures (introduced by Krupinski in [9]). Using the Haar measure of locally compact groups, we introduce an independence, called μ-independence, in μ-structures having good properties. With this independence notion, we develop geometric stability theory for μ-structures. Then we see some structural theorems for compact groups which are μ-structure. We also give examples of profinite structures where μ-independence is different from nm-independence introduced by Krupinski for Polish structures.In an appendix, Cohen and Wesolek show that a profinite branch group gives a small action on the boundary of a rooted tree so that this actions provides a small profinite structure on the boundary of a rooted tree.     

Global square sequences in extender models

We present a construction of a global square sequence in extender models with λ-indexing.     

On a combinatorial property of choice functions of finite sets

We consider choice functions k[X]→X, where X is a finite set and k[X] denotes the set of all k-subsets of X. We define a property of domination for such maps generalizing the classical case k=2 (tournaments) and prove the existence of a dominating element generalizing the existence of a 2-root (king) in the classical case.     

On the Burkard-Hammer condition for hamiltonian split graphs

A graph G=(V,E) is called a split graph if there exists a partition V=I∪K such that the subgraphs of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary but not sufficient condition for hamiltonian split graphs with |I|<|K|. In this paper, we show that the Burkard-Hammer condition is also sufficient for the existence of a Hamilton cycle in a split graph G such that 5≠|I|<|K| and the minimum degree δ(G)?|I|-3. For the case 5=|I|<|K|, all split graphs satisfying the Burkard-Hammer condition but having no Hamilton cycles are also described.     

Compact spaces, elementary submodels, and the countable chain condition, II

Given a space 〈X,T〉 in an elementary submodel of H(θ), define XM to be X∩M with the topology generated by . It is established that if XM is compact and satisfies the countable chain condition, while X is not scattered and has cardinality less than the first inaccessible cardinal, then X=XM. If the character of XM is a member of M, then “inaccessible” may be replaced by “1-extendible”.     

QN-spaces, wQN-spaces and covering properties

The main results of the paper are as follows: covering characterizations of wQN-spaces, covering characterizations of QN-spaces and a theorem saying that Cp(X) has the Arkhangel'ski?ˇ property (α₁) provided that X is a QN-space. The latter statement solves a problem posed by M. Scheepers [M. Scheepers, Cp(X) and Arhangel'ski?ˇ's αi-spaces, Topology Appl. 89 (1998) 265-275] and for Tychonoff spaces was independently proved by M. Sakai [M. Sakai, The sequence selection properties of Cp(X), Preprint, April 25, 2006]. As the most interesting result we consider the equivalence that a normal topological space X is a wQN-space if and only if X has the property S₁(Γshr,Γ). Moreover we show that X is a QN-space if and only if Cp(X) has the property (α₀), and for perfectly normal spaces, if and only if X has the covering property (β₃).     

Fusion and large cardinal preservation

In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ?κ   not only does not collapse κ⁺${\kappa }^{+}$ but also preserves the strength of κ (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf. Dobrinen and Friedman (2010) [3], Friedman and Halilovi? (2011) [5], Friedman and Honzik (2008) [6], Friedman and Magidor (2009) [8], Friedman and Zdomskyy (2010) [10], Honzik (2010) [12]).     

Projective mad families

Using almost disjoint coding we prove the consistency of the existence of a definable ω-mad family of infinite subsets of ω (resp. functions from ω to ω) together with b=2^ω=ω₂.     

Model theoretic dynamics in Galois fashion

We fix a monster model $\mathfrak{D}$ of some stable theory and investigate substructures of $\mathfrak{D}$ which are existentially closed within the class of substructures equipped with an action of a fixed group G. We describe them as PAC substructures of $\mathfrak{D}$ and obtain results related to Galois theory.Assuming that the class of these existentially closed substructures is elementary, we show that, under the assumption of having bounded models, its theory is simple and eliminates quantifiers up to some existential formulas. Moreover, this theory codes finite sets and allows a geometric elimination of imaginaries, but not always a weak elimination of imaginaries.