Indestructibility and destructible measurable cardinals

Say that \({\kappa}\)’s measurability is destructible if there exists a < \({\kappa}\)-closed forcing adding a new subset of \({\kappa}\) which destroys \({\kappa}\)’s measurability. For any δ, let λ_δ =_df The least beth fixed point above δ. Suppose that \({\kappa}\) is indestructibly supercompact and there is a measurable cardinal λ > \({\kappa}\). It then follows that \({A_{1} = \{\delta < \kappa \mid \delta}\) is measurable, δ is not a limit of measurable cardinals, δ is not δ⁺ strongly compact, and δ’s measurability is destructible when forcing with partial orderings having rank below λ_δ} is unbounded in \({\kappa}\). On the other hand, under the same hypotheses, \({A_{2} = \{\delta < \kappa \mid \delta}\) is measurable, δ is not a limit of measurable cardinals, δ is not δ⁺ strongly compact, and δ′s measurability is indestructible when forcing with either Add(δ, 1) or Add(δ, δ⁺)} is unbounded in \({\kappa}\) as well. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two distinct models in which either \({A_{1} = \emptyset}\) or \({A_{2} = \emptyset}\). In each of these models, both of which have restricted large cardinal structures above \({\kappa}\), every measurable cardinal δ which is not a limit of measurable cardinals is δ⁺ strongly compact, and there is an indestructibly supercompact cardinal \({\kappa}\). In the model in which \({A_{1} = \emptyset}\), every measurable cardinal δ which is not a limit of measurable cardinals is <λ_δ strongly compact and has its <λ_δ strong compactness (and hence also its measurability) indestructible when forcing with δ-directed closed partial orderings having rank below λ_δ. The choice of the least beth fixed point above δ is arbitrary, and other values of λ_δ are also possible.     

More about λ-support iterations of (<λ)-complete forcing notions

This article continues Ros?anowski and Shelah (Int J Math Math Sci 28:63–82, 2001; Quaderni di Matematica 17:195–239, 2006; Israel J Math 159:109–174, 2007; 2011; Notre Dame J Formal Logic 52:113–147, 2011) and we introduce here a new property of (<λ)-strategically complete forcing notions which implies that their λ-support iterations do not collapse λ ⁺ (for a strongly inaccessible cardinal λ).     

Martin’s Maximum and tower forcing

There are several examples in the literature showing that compactness-like properties of a cardinal κ cause poor behavior of some generic ultrapowers which have critical point κ (Burke [1] when κ is a supercompact cardinal; Foreman-Magidor [6] when κ = ω ₂ in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $\overrightarrow I $ is a tower of ideals which concentrates on the class $GI{C_{{\omega _1}}}$ of ω ₁-guessing, internally club sets, then $\overrightarrow I $ is not presaturated (a set is ω ₁-guessing iff its transitive collapse has the ω ₁-approximation property as defined in Hamkins [10]). This theorem, combined with work from [16], shows that if PFA ⁺ or MM holds and there is an inaccessible cardinal, then there is a tower with critical point ω ₂ which is not presaturated; moreover, this tower is significantly different from the non-presaturated tower already known (by Foreman-Magidor [6]) to exist in all models of Martin’s Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at ω ₂ has similar implications for towers of ideals which concentrate on the wider class $GI{C_{{\omega _1}}}$ of ω ₁-guessing, internally stationary sets. Finally, we show that the word “presaturated” cannot be replaced by “precipitous” in the theorems above: Martin’s Maximum (which implies SRP and the Tree Property at ω ₂) is consistent with a precipitous tower on $GI{C_{{\omega _1}}}$ .     

Weak square and stationary reflection

It is well-known that the square principle \({\square_\lambda}\) entails the existence of a non-reflecting stationary subset of λ⁺, whereas the weak square principle \({\square^{*} _\lambda}\) does not. Here we show that if μ^cf(λ) < λ for all μ < λ, then \({\square^{*} _\lambda}\) entails the existence of a non-reflecting stationary subset of \({E^{\lambda^+}_{{\rm cf}(\lambda)}}\)in the forcing extension for adding a single Cohen subset of λ⁺.It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplete forcing axiom (SCFA), proving that SCFA entails the failure of \({\square^{*} _\lambda}\) for every singular cardinal λ of countable cofinality.     

Indestructibility, instances of strong compactness, and level by level inequivalence

Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ ⁺ strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which ${A = \emptyset}$ . The first of these contains a supercompact cardinal κ and is such that no cardinal δ > κ is measurable, κ’s supercompactness is indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing, and every measurable cardinal δ < κ is δ ⁺ strongly compact. The second of these contains a strong cardinal κ and is such that no cardinal δ > κ is measurable, κ’s strongness is indestructible under < κ-strategically closed, (κ ⁺, ∞)-distributive forcing, and level by level inequivalence between strong compactness and supercompactness holds. The model from the first of our forcing constructions is used to show that it is consistent, relative to a supercompact cardinal, for the least cardinal κ which is both strong and has its strongness indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing to be the same as the least supercompact cardinal, which has its supercompactness indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing. It further follows as a corollary of the first of our forcing constructions that it is possible to build a model containing a supercompact cardinal κ in which no cardinal δ > κ is measurable, κ is indestructibly supercompact, and every measurable cardinal δ < κ which is not a limit of measurable cardinals is δ ⁺ strongly compact.     

Ein zerlegungssatz für P(ϰ)

We shall prove the following partition theorems:

1. For every setS and for each cardinal ? ≥ ω, |S| ≥ ? there exists a partitionT: [S]^? → 2^? such that for every pairwise disjoint familie and everyα < 2^? there exists a set
2. Suppose ? ≥ ω, 2 ^{andS an arbitrary set, 2|S| ≤ (2?)+ω Then there exists a partitionT: P(S) → 2? such that for every pairwise disjoint family and everyα < 2? there exists a set Both theorems will give partial answers to an Erd?s problem.}

     

Noetherian type in topological products

The cardinal invariant Noetherian type Nt(X) of a topological space X was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2:

1. There are spaces X and Y such that Nt(X×Y)< min{Nt(X), Nt(Y)}.
2. In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace.

The Noetherian type of the Cantor Cube of weight \({\aleph _\omega }\) with the countable box topology, \({({2^{{\aleph _\omega }}})_\delta }\) , is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of \({\aleph _\omega }\) . We discuss the influence of principles like \({\square _{{\aleph _\omega }}}\) and Chang’s conjecture for \({\aleph _\omega }\) on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms). Within PCF theory we establish the existence of an (?₄, ?₁)-sparse covering family of countable subsets of \({\aleph _\omega }\) (Theorem 3.20). From this follows an absolute upper bound of ?₄ on the Noetherian type of \({({2^{{\aleph _\omega }}})_\delta }\) . The proof uses a method that was introduced by Shelah in 1993 [33].     

Sine, cosine transforms and classical function classes

We study the continuity and smoothness properties of functions f ∈ L ¹([0, ∞)) whose sine transforms $ \hat f_s $ and cosine tranforms $ \hat f_c $ belong to L ¹([0,∞)). We give best possible sufficient conditions in terms of $ \hat f_s $ and $ \hat f_c $ to ensure that f belongs to one of the Lipschitz classes Lip α and lip α for some 0 < α ≤ 1, or to one of the Zygmund classes Zyg α and zyg α for some 0 < α ≤ 2. The conditions given by us are not only sufficient, but also necessary in the case when the sine and cosine transforms are nonnegative. Our theorems are extensions of the corresponding theorems by Boas from sine and cosine series to sine and cosine transforms.     

A base-matrix lemma for sets of rationals modulo nowhere dense sets

We study some properties of the quotient forcing notions ${Q_{tr(I)} = \wp(2^{< \omega})/tr(I)}$ and P _I ?= B(2 ^ω )/I in two special cases: when I is the σ-ideal of meager sets or the σ-ideal of null sets on 2 ^ω . We show that the remainder forcing R _I =?Q _tr(I)/P _I is σ-closed in these cases. We also study the cardinal invariant of the continuum ${\mathfrak{h}_{\mathbb{Q}}}$ , the distributivity number of the quotient ${Dense(\mathbb{Q})/nwd}$ , in order to show that ${\wp(\mathbb{Q})/nwd}$ collapses ${\mathfrak{c}}$ to ${\mathfrak{h}_{\mathbb{Q}}}$ , thus answering a question addressed in Balcar et?al. (Fundamenta Mathematicae 183:59–80, 2004).     

Easton’s theorem in the presence of Woodin cardinals

Under the assumption that δ is a Woodin cardinal and GCH holds, I show that if F is any class function from the regular cardinals to the cardinals such that (1) ${\kappa < {\rm cf}(F(\kappa))}$ , (2) ${\kappa < \lambda}$ implies ${F(\kappa) \leq F(\lambda)}$ , and (3) δ is closed under F, then there is a cofinality-preserving forcing extension in which 2 ^γ = F(γ) for each regular cardinal γ < δ, and in which δ remains Woodin. Unlike the analogous results for supercompact cardinals [Menas in Trans Am Math Soc 223:61–91, (1976)] and strong cardinals [Friedman and Honzik in Ann Pure Appl Logic 154(3):191–208, (2008)], there is no requirement that the function F be locally definable. I deduce a global version of the above result: Assuming GCH, if F is a function satisfying (1) and (2) above, and C is a class of Woodin cardinals, each of which is closed under F, then there is a cofinality-preserving forcing extension in which 2 ^γ = F(γ) for all regular cardinals γ and each cardinal in C remains Woodin.     

Multivariate cardinal interpolation with radial-basis functions

For a radial-basis function?∶?→? we consider interpolation on an infinite regular lattice , tof∶? ⁿ→?, whereh is the spacing between lattice points and the cardinal function , satisfiesX(j)=δ _oj for allj∈? ⁿ. We prove existence and uniqueness of such cardinal functionsX, and we establish polynomial precision properties ofI _h for a class of radial-basis functions which includes \(\varphi (r) = r^{2q + 1} \) , \(\varphi (r) = r^{2q} \log r,\varphi (r) = \sqrt {r^2 + c^2 } \) , and \(\varphi (r) = 1/\sqrt {r^2 + c^2 } \) whereq∈? ₊. We also deduce convergence orders ofI _hf to sufficiently differentiable functionsf whenh→0.     

Non-saturation of the nonstationary ideal on P κ (λ) in case κ ≤ cf (λ) < λ

Given a regular cardinal κ > ω ₁ and a cardinal λ with κ?≤ cf (λ) < λ, we show that NS _κ,λ | T is not λ⁺-saturated, where T is the set of all ${a\in P_\kappa (\lambda)}$ such that ${| a | = | a \cap \kappa|}$ and ${{\rm cf} \big( {\rm sup} (a\cap\kappa)\big) = {\rm cf} \big({\rm sup} (a)\big) = \omega}$ .     

Principal eigenvalue of the fractional Laplacian with a large incompressible drift

We study the principal Dirichlet eigenvalue of the operator \({L_A=\Delta^{\alpha/2}+Ab(x)\cdot\nabla}\) , on a bounded C ^1,1 regular domain D. Here \({\alpha\in(1,2)}\) , \({\Delta^{\alpha/2}}\) is the fractional Laplacian, \({A\in\mathbb{R}}\) , and b is a bounded d-dimensional divergence-free vector field in the Sobolev space W ^1,2d/(d+α)(D). We prove that the eigenvalue remains bounded, as A→ + ∞, if and only if b has non-trivial first integrals in the domain of the quadratic form of \({\Delta^{\alpha/2}}\) for the Dirichlet condition.     

Cichoń’s diagram,regularity properties and {\varvec{\Delta}^1_3} sets of reals

We study regularity properties related to Cohen, random, Laver, Miller and Sacks forcing, for sets of real numbers on the \({\varvec{\Delta}^1_3}\) level of the projective hieararchy. For \({\varvec{\Delta}^1_2}\) and \({\varvec{\Sigma}^1_2}\) sets, the relationships between these properties follows the pattern of the well-known Cichoń diagram for cardinal characteristics of the continuum. It is known that assuming suitable large cardinals, the same relationships lift to higher projective levels, but the questions become more challenging without such assumptions. Consequently, all our results are proved on the basis of ZFC alone or ZFC with an inaccessible cardinal. We also prove partial results concerning \({\varvec{\Sigma}^1_3}\) and \({\varvec{\Delta}^1_4}\) sets.     

Uniqueness of non-linear ground states for fractional Laplacians in {\mathbb{R}}

We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation $$(-\Delta)^s Q+Q-Q^{\alpha+1}= 0 \quad {\rm in} \, \mathbb{R},$$ ( ? Δ ) s Q + Q ? Q α + 1 = 0 i n R , where 0 < s < 1 and 0 < α < 4s/(1?2s) for ${s<\frac{1}{2}}$ s < 1 2 and 0 < α < ∞ for ${s\geq \frac{1}{2}}$ s ≥ 1 2 . Here (?Δ) ^s denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for ${s=\frac{1}{2}}$ s = 1 2 and α = 1 in [5] for the Benjamin–Ono equation. As a technical key result in this paper, we show that the associated linearized operator L ₊ = (?Δ) ^s +1?(α+1)Q ^α is non-degenerate; i.e., its kernel satisfies ker L ₊ = span{Q′}. This result about L ₊ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin–Ono (BO) and Benjamin–Bona–Mahony (BBM) water wave equations.     

Many different covering numbers of Yorioka’s ideals

For ${b \in {^{\omega}}{\omega}}$ , let ${\mathfrak{c}^{\exists}_{b, 1}}$ be the minimal number of functions (or slaloms with width 1) to catch every functions below b in infinitely many positions. In this paper, by using the technique of forcing, we construct a generic model in which there are many coefficients ${\mathfrak{c}^{\exists}_{{b_\alpha}, 1}}$ with pairwise different values. In particular, under the assumption that a weakly inaccessible cardinal exists, we can construct a generic model in which there are continuum many coefficients ${\mathfrak{c}^{\exists}_{{b_\alpha}, 1}}$ with pairwise different values. In conjunction with these results, we give a generic model in which there are many Yorioka’s ideals ${\mathcal{I}_{f_\alpha}}$ with pairwise different covering numbers.     

Triangulations of the sphere,bitrades and abelian groups

Let \(\mathcal{G}\) be a triangulation of the sphere with vertex set V, such that the faces of the triangulation are properly coloured black and white. Motivated by applications in the theory of bitrades, Cavenagh and Wanless defined \(\mathcal{A}_W\) to be the abelian group generated by the set V, with relations r+c+s = 0 for all white triangles with vertices r, c and s. The group \(\mathcal{A}_B\) can be de fined similarly, using black triangles. The paper shows that \(\mathcal{A}_W\) and \(\mathcal{A}_B\) are isomorphic, thus establishing the truth of a well-known conjecture of Cavenagh and Wanless. Connections are made between the structure of \(\mathcal{A}_W\) and the theory of asymmetric Laplacians of finite directed graphs, and weaker results for orientable surfaces of higher genus are given. The relevance of the group \(\mathcal{A}_W\) to the understanding of the embeddings of a partial latin square in an abelian group is also explained.